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Introduction to Best-Packing and Best-Covering

  • Sergiy V. BorodachovEmail author
  • Douglas P. Hardin
  • Edward B. Saff
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Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we discuss two fundamental problems of discrete geometry, the best-packing problem and the best-covering problem. In Section 3.1 the general best-packing problem is introduced. We show that it is the limiting case as \(s\rightarrow \infty \) of the Riesz s-energy problem, see Proposition 3.1.2. In that section we also estimate the minimal pairwise separation of an N-point s-energy minimizing configuration on a path connected compact set. Section 3.2 introduces the general best-covering problem and discusses the basic relationship between best-packing distance and mesh ratio on a given compact set.

In this chapter we discuss two fundamental problems of discrete geometry, the best-packing problem and the best-covering problem. In Section 3.1 the general best-packing problem is introduced. We show that it is the limiting case as \(s\rightarrow \infty \) of the Riesz s-energy problem, see Proposition 3.1.2. In that section we also estimate the minimal pairwise separation of an N-point s-energy minimizing configuration on a path connected compact set. Section 3.2 introduces the general best-covering problem and discusses the basic relationship between best-packing distance and mesh ratio on a given compact set. Section 3.3 presents exact solutions to best-packing and best-covering problems on the sphere for certain small cardinalities of configurations. In particular, the optimality of the regular simplex inscribed in \(S^d\) is shown as well as the optimality of the regular octahedron and icosahedron on \(S^2\). The problem of the densest packing of equal non-overlapping circles in the plane and the problem of the most economical covering of the plane by equal circles are discussed in Section 3.4. Their solutions are used in Section 3.7 to obtain the leading term as \(N\rightarrow \infty \) of the N-point best-packing distance and of the minimal N-point covering radius of the sphere \(S^2\) in \(\mathbb R^3\).

As usual, in the last section we provide notes and historical references for the theorems and related results discussed in the chapter.

3.1 Best-Packing and Its Relation to Minimal Energy

In this section, we introduce a discrete geometric problem known as the best-packing problem and state it in a general metric space. We define the separation distance  of an N-point configuration \(\omega _N=\{x_1,\ldots , x_N\}\) in a metric space \((A,\rho )\) as
$$\begin{aligned} \delta ^\rho (\omega _N):=\min \limits _{1\le i\ne j\le N}{\rho (x_i, x_j)}. \end{aligned}$$
(3.1.1)

Definition 3.1.1

The N-point best-packing distance  on the set A is defined as
$$\begin{aligned} \delta ^\rho _N(A):=\sup \{\delta ^\rho (\omega _N) : \omega _N\subset A \}, \end{aligned}$$
(3.1.2)
and best-packing configurations  are N-point configurations \(\omega ^*_N\subset A\) attaining the supremum in (3.1.2). When \(A\subset \mathbb R^d\) and \(\rho \) is the Euclidean distance, we omit the superscript \(\rho \) in definitions (3.1.1) and (3.1.2).

The problem of finding the best-packing distance and best-packing configurations on a given set is called the best-packing problem (or sphere packing problem ) . In other terms, one is looking for the supremum of the common radius of N pairwise disjoint open balls centered at points of the set A.

The separation distance \(\delta ^\rho (\omega _N)\) is a continuous function of \(\omega _N\) in the topology on the Nth Cartesian power of A generated by the metric \(\rho \). If \((A,\rho )\) is a compact metric space, best-packing configurations exist and the supremum in (3.1.2) can be replaced by a maximum.

For \(N=2\), one trivially has \(\delta _2^\rho (A)={{\,\mathrm{diam}\,}}A\). If A is a subset of the Euclidean space \(\mathbb R^p\), we will omit the superscript \(\rho \) in the notation of the separation and best-packing distance. One can also easily verify that, for example, \(\delta _N([0,1])=\frac{1}{N-1}\) and that \(\delta _N(S^1)=2\sin \frac{\pi }{N}\), \(N\ge 2\).

The best-packing problem on the sphere is known as the Tammes problem . Some fundamental results on this problem are discussed in Section 3.3 for the sphere \(S^2\) and in Sections  5.7 and  5.9 for higher dimensional spheres. We also address this problem in other parts of the book: Section 3.4 discusses the problem of the densest packing of equal circles in the Euclidean plane and Chapter  13 deals with the asymptotic behavior of the N-point best-packing distance and the weak\(^*\) limit distribution of best-packing N-point configurations on rectifiable sets in \(\mathbb R^p\) as \(N\rightarrow \infty \).

The best-packing problem turns out to be the limiting case of the minimal Riesz s-energy problem when s gets large, as we next show. We denote by \(\omega ^s_N=\{x_1^s,\ldots , x_N^s\}\) an s-energy minimizing configuration on A if \(0<s<\infty \) and a best-packing configuration on A if \(s=\infty \). With regard to the extended real number limits in \([0,\infty ]\), we agree that \(1/0=0^{-s}=\infty ^{s}=\infty \), \(1/\infty =\infty ^{-s}=0\), \(s>0\).

Proposition 3.1.2

Let \(N\in \mathbb N\), \(N\ge 2\), be fixed and \((A,\rho )\) be an infinite compact metric space. Then
$$\begin{aligned} \lim \limits _{s\rightarrow \infty }{\mathcal E_s(A, N)^{1/s}}=\frac{1}{\delta ^\rho _N(A)}. \end{aligned}$$
(3.1.3)
Moreover, every cluster configuration (in the N-multi-set topology1) as \(s\rightarrow \infty \) of a set \(\{\omega ^s_N\}_{s>0}\) of s-energy minimizing N-point configurations on A is an N-point best-packing configuration \(\omega _N^\infty \) on A and
$$\begin{aligned} \lim \limits _{s\rightarrow \infty }\delta ^\rho (\omega _N^s)=\delta ^\rho (\omega _N^\infty )=\delta _N^\rho (A). \end{aligned}$$
(3.1.4)

Proof

Note that
$$\begin{aligned} \mathcal E_s(A, N)^{1/s}=E_s(\omega ^s_N)^{1/s}\ge \frac{1}{\delta ^\rho (\omega ^s_N)}\ge \frac{1}{\delta ^\rho _N(A)}. \end{aligned}$$
(3.1.5)
On the other hand,
$$ {\mathcal E_s(A, N)^{1/s}}\le {E_s(\omega _N^\infty )^{1/s}} $$
$$\begin{aligned} =\frac{1}{\delta ^\rho (\omega _N^\infty )}\left( \sum \limits _{1\le i\ne j\le N}{\left( \frac{\delta ^\rho (\omega _N^\infty )}{\rho (x_i^\infty , x_j^\infty )}\right) ^s}\right) ^{1/s}\le \frac{1}{\delta ^\rho (\omega _N^\infty )}(N(N-1))^{1/s}. \end{aligned}$$
(3.1.6)
Thus,
$$ \limsup \limits _{s\rightarrow \infty }{\mathcal E_s(A, N)^{1/s}}\le \frac{1}{\delta ^\rho (\omega _N^\infty )}=\frac{1}{\delta ^\rho _N(A)}, $$
and taking into account (3.1.5), we obtain (3.1.3).
Fig. 3.1

The behavior of the minimal pairwise separation of computed configurations of \(N=500\) points on \(S^2\) that minimize the s-energy for different values of s. Note the apparent monotonic behavior of the separation distance. The continuous horizontal line corresponds to the (computed) best-packing distance of a 500-point configuration on \(S^2\)

Now let \(\omega _N\) be any cluster point of the sequence \(\{\omega ^s_N\}_{s>0}\) as \(s\rightarrow \infty \); i.e., there is some sequence \(\{s_k\}_{k=1}^{\infty }\) of positive numbers such that \(s_k\rightarrow \infty \) and
$$ \lim \limits _{k\rightarrow \infty }{\omega _N^{s_k}}=\omega _N. $$
Then from (3.1.5) and (3.1.6) we have
$$\begin{aligned} \frac{1}{\delta ^\rho (\omega ^{s_k}_N)}\le E_{s_k}(\omega ^{s_k}_N)^{1/{s_k}}=\mathcal E_{s_k}(A, N)^{1/{s_k}}\le \frac{1}{\delta ^\rho (\omega ^\infty _N)}(N(N-1))^{1/{s_k}}. \end{aligned}$$
(3.1.7)
Note that \(\delta ^\rho (\omega ^{s_k}_N)\) is eventually bounded away from zero, which implies that \(\omega _N\) has N distinct points. Letting \(k\rightarrow \infty \) in (3.1.7), we obtain that \(1/\delta ^\rho (\omega _N)\le 1/\delta ^\rho (\omega ^\infty _N)\). Hence, \(\delta ^\rho (\omega _N)\ge \delta ^\rho (\omega ^\infty _N)\); i.e., \(\omega _N\) is a best-packing configuration. Relation (3.1.4) now follows from relation (3.1.7) and the fact that \(\delta ^\rho (\omega _N^{s_k})\le \delta ^\rho (\omega _N^\infty )\).    \(\square \)

Figure 3.1 illustrates for increasing values of s the behavior of the separation distance for computed s-energy minimizing 500-point configurations on \(S^2\) along with the best packing distance for 500 points; i.e., \(\delta (\omega _{500}^\infty )\).

Next we establish a basic estimate for the minimal pairwise distance between points in optimal configurations on path connected2 spaces.

Proposition 3.1.3

Let \(s>0\), \((A,\rho )\) be a path connected compact metric space and \(\omega _N^s\), \(N\ge 2\), be an s-energy minimizing N-point configuration on A. Then
$$\begin{aligned} \delta _N^\rho (A)\ge \delta ^\rho (\omega _N^s)\ge \frac{1}{2} \left( \frac{N}{2\mathcal E_s(A, N)}\right) ^{1/s}. \end{aligned}$$
(3.1.8)

Proof

If A consists of one point, then inequality (3.1.8) holds trivially for \(N\ge 2\). Assume that A contains two or more distinct points (then A contains a subset of cardinality continuum since A is path connected). Denote by \(x_1,\ldots , x_N\) the points in \(\omega _N^s\) and let
$$ U_i:=\sum \limits _{j: j\ne i}{\frac{1}{\rho (x_i, x_j)^s}}. $$
Let \(1\le k\le N\) be an index such that \(U_k=\min \{U_i: 1\le i\le N\}\) and let \(x_l\ne x_k\) be a point in \(\omega _N^s\) closest to \(x_k\). Since A is path connected, there is a continuous curve \(\gamma :[0,1]\rightarrow A\) that joins points \(x_k\) and \(x_l\) (\(\gamma (0)=x_k\) and \(\gamma (1)=x_l\)). Since the function \(g(t):=\rho (x_k,\gamma (t))\) is continuous on [0, 1], there is a point \(z=\gamma (t_0)\in A\) such that \(g(t_0)=\rho (x_k,z)=\rho (x_k, x_l)/2\).
For any \(i\ne k\) we have
$$ \rho (z,x_i)\ge \rho (x_k,x_i)-\rho (z,x_k)=\rho (x_k, x_i)-\frac{1}{2}\rho (x_k, x_l)\ge \frac{1}{2}\rho (x_k, x_i). $$
Hence,
$$ U(z):=\sum \limits _{i=1}^{N}\frac{1}{\rho (z, x_i)^s}=\frac{1}{\rho (z, x_k)^s}+\sum \limits _{i : i\ne k}\frac{1}{\rho (z, x_i)^s} $$
$$ \le \frac{2^s}{\rho (x_k, x_l)^s}+2^s\sum \limits _{i : i\ne k}\frac{1}{\rho (x_k, x_i)^s}\le 2^{s+1}U_k. $$
Let \(1\le m\le N\), \(m\ne k\), be any index. Then
$$ \mathcal E_s(A, N)\le E_s\left( \left( \omega _N^s\setminus \{x_m\}\right) \cup \{z\}\right) $$
$$ \le E_s(\omega _N^s)-2U_m+2U(z)\le \mathcal E_s(A, N)-2U_m+2^{s+2}U_k. $$
For every \(1\le m\le N\) (including \(m=k\)), we then obtain
$$ U_m\le 2^{s+1}U_k\le \frac{2^{s+1}}{N}\sum \limits _{i=1}^{N}{U_i}=\frac{2^{s+1}}{N}E_s(\omega _N^s)=\frac{2^{s+1}}{N}\mathcal E_s(A, N). $$
If \(x_i, x_j\), \(i\ne j\), are any two points from \(\omega _N^s\), we have
$$ \frac{1}{\rho (x_i, x_j)^s}\le U_i\le \frac{2^{s+1}}{N}\mathcal E_s(A, N), $$
which implies that
$$ \rho (x_i, x_j)\ge \frac{1}{2} \left( \frac{N}{2\mathcal E_s(A, N)}\right) ^{1/s},\ \ i\ne j, $$
and (3.1.8) follows.    \(\square \)

The estimate of Proposition 3.1.3 is especially useful when sharp asymptotics is known for the minimal energy \(\mathcal E_s(A, N)\) as \(N\rightarrow \infty \), as will be the case for rectifiable curves (see Theorem  8.2.3) and for the sphere \(S^d\) when \(s>d\). Indeed, in the latter case \(\mathcal E_s(S^d, N)\) has asymptotic order \(N^{1+s/d}\) so that (3.1.8) implies that for some constant \(c_s>0\), \(\delta (\omega _N^s)\ge c_s/N^{1/d}\), for \(N\ge 2\), which is the optimal separation order of a sequence of N-point configurations on \(S^d\).

3.2 The Covering Problem and Its Relation to Packing

Here we will introduce one more fundamental problem in discrete geometry, which is somewhat dual to that of best-packing.

Definition 3.2.1

Let A be a non-empty set in a metric space \((X,\rho )\). The covering radius  (also called the mesh norm  or the fill radius ) of a configuration \(\omega _N=\{x_1,\ldots , x_N\}\subset X\) with respect to the set A is defined as
$$\begin{aligned} \eta (\omega _N, A)=\sup \limits _{x\in A}{\min \limits _{i= {1,\ldots , N}}\rho (x, x_i)}. \end{aligned}$$
(3.2.1)

The covering radius of \(\omega _N\) with respect to A can be interpreted as the minimal radius of N closed balls (of equal radii) centered at the points in \(\omega _N\) whose union contains the set A. One can also think of the quantity \(\eta (\omega _N, A)\) as the supremum of the radius of an open ball centered at a point in A that does not contain any point from \(\omega _N\). In approximation theory this quantity is known as the best approximation of the set A by the configuration \(\omega _N\).

The most general setting of the best covering problem is as follows. Let A and D be non-empty subsets of a metric space \((X,\rho )\). The minimal N-point covering radius of the set A relative to the set D is defined as
$$\begin{aligned} \eta _N^D(A):=\inf \limits \{\eta (\omega _N, A) : \omega _N\subset D\}. \end{aligned}$$
(3.2.2)
The quantity \(\eta _N^D(A)\) equals the infimum of the common radius of N balls centered at points of D whose union covers the set A. Clearly, if \(D_1\subset D_2\), then \(\eta _{N}^{D_1}(A)\ge \eta _N^{D_2}(A)\) and if \(A_1\subset A_2\), then \(\eta _N^D(A_1)\le \eta _N^{D}(A_2)\). As in the case of minimal s-energy and best-packing, the best-covering problem is the limiting case of the two-plate s-polarization problem as \(s\rightarrow \infty \) (see Section  14.4).
We are mainly interested in the case \(A=D\); i.e., when the centers of the balls covering A are chosen from the same set A. We will compare this case with the case \(D=X\); i.e., when no restriction is placed on the location of the centers of balls that cover A. Concerning the case \(A=D\), we define the minimal N-point covering radius of a set A as
$$\begin{aligned} \eta _N(A):=\eta _N^A(A)=\inf \limits \{\eta (\omega _N, A) : \omega _N\subset A\}. \end{aligned}$$
(3.2.3)
A configuration \(\omega _N^c\) that attains the infimum on the right-hand side of (3.2.3) is called an N-point best-covering configuration for A .
When A is a non-empty subset of \(\mathbb R^p\) it is easy to show that the quantity (3.2.3) posesses the following properties:
  1. (i)

    \(\eta _N(\alpha A)=\alpha \eta _N(A)\) for every \(\alpha >0\);

     
  2. (ii)

    if A is unbounded, then \(\eta _N(A)=\infty \);

     
  3. (iii)

    for the closure \(\overline{A}\) of the set A, there holds \(\eta _N(\overline{A})=\eta _N(A)\).

     

In contrast to best-packing, if \(A_1\subset A_2\), the inequality \(\eta _N(A_1)\le \eta _N(A_2)\) may fail. For example, if \(N=2\), \(A_1=S^1\), and \(A_2=S^1\cup \{(0,0)\}\), we have \(\eta _2(A_1)=\sqrt{2}\) and \(\eta _2(A_2)=1\). However, we emphasize one case when the monotonicity property still holds.

Proposition 3.2.2

If \( A\subset B\subset \mathbb R^p\) is a non-empty convex set, \(\eta _N(A)\le \eta _N(B)\).

As we mentioned above, if \(A\subset \mathbb R^p\), one has \(\eta _N(A)=\eta _N^A(A)\ge \eta _N^{\mathbb R^p}(A)\) and the inequality may be strict. For example, if \(N=2\), \(p=2\), and \(A=S^1\), we have \(\eta _2(A)=\sqrt{2}\) while \(\eta _2^{\mathbb R^2}(A)=1\). One case when we still have equality is the described below.

Proposition 3.2.3

If \(A\subset \mathbb R^p\) is a non-empty convex set, \(\eta _N^{\mathbb R^p}(A)=\eta _N(A)\).

Proof of Propositions 3.2.2 and 3.2.3 Let \(\omega _N:=\{x_1,\ldots , x_N\}\) be an arbitrary configuration in \(\mathbb R^p\) and \(\varphi :\mathbb R^p\rightarrow \overline{A}\) be the mapping such that \(\varphi (x)\) is the point in the closure \(\overline{A}\) of A that is the closest to x. In view of the convexity of \(\overline{A}\), for any points \(x\in \mathbb R^p\) and \(y\in \overline{A}\), we have \(\left| x-y\right| \ge \left| \varphi (x)-y\right| \) (see Proposition A.1.1 in the Appendix). Then
$$\begin{aligned} \begin{aligned} \eta (\omega _N,\overline{A})&=\sup \limits _{y\in \overline{A}}{\min \limits _{i=1,\dots , N}{\left| y-x_i\right| }}\ge \sup \limits _{y\in \overline{A}}{\min \limits _{i=1,\dots , N}{\left| y-\varphi (x_i)\right| }}\\&=\eta (\varphi (\omega _N),\overline{A})\ge \inf \limits _{\omega \subset \overline{A}\atop \#\omega =N}\eta (\omega ,\overline{A})=\eta _N(\overline{A}). \end{aligned} \end{aligned}$$
(3.2.4)
In view of arbitrariness of the configuration \(\omega _N\subset \mathbb R^p\), we obtain that \(\eta _N^{\mathbb R^p}(A)=\eta _N^{\mathbb R^p}(\overline{A})\ge \eta _N(\overline{A})=\eta _N(A)\). In view of the monotonicity of the quantity \(\eta _N^D(A)\) with respect to the set D, we obtain that \(\eta _N^{\mathbb R^p}(A)\le \eta _N(A)\), and hence, \(\eta _N^{\mathbb R^p}(A)=\eta _N(A)\), which proves Proposition 3.2.3.
To prove Proposition 3.2.2 we choose \(\omega _N\) from the set B. Then (3.2.4) and Proposition 3.2.3 imply that
$$ \eta (\omega _N,\overline{B})\ge \eta (\omega _N,\overline{A})\ge \eta _N^{\mathbb R^p}(\overline{A})=\eta _N^{\mathbb R^p}(A)=\eta _N(A). $$
In view of property (iii) above and the arbitrariness of the configuration \(\omega _N\subset B\), we obtain the inequality \(\eta _N(B)=\eta _N(\overline{B})\ge \eta _N(A)\).    \(\square \)
Next we present several examples when the best covering problem has an exact solution. In the case \(N=1\), a configuration \(\omega _1=\{x\}\) is a 1-point best-covering configuration on the set A if and only if x is a Chebyshev center of A. We recall that a point \(x\in A\) is a Chebyshev center  of A if
$$ \sup \limits _{y\in A}\rho (x,y)\le \sup \limits _{y\in A}\rho (z, y) $$
for any point \(z\in A\). If x is a Chebyshev center of A, then the quantity \(\eta _1(A)=\sup \{\rho (x, y) : y\in A\}\) is called the (Chebyshev) radius  of the set A.

It is not difficult to verify the following statement concerning the best-covering problem for an interval and a circle.

Proposition 3.2.4

For the interval [0, 1], the set \(\omega _N^c:=\left\{ \frac{2k-1}{2N}\right\} _{k=1}^{N}\), \(N\in \mathbb N\), is the unique N-point best-covering configuration, and \(\eta _N([0,1])=\frac{1}{2N}\). For the unit circle \(S^1\), any configuration of N equally spaced points on \(S^1\) is a best-covering configuration on \(S^1\) and these are the only N-point best-coverings on \(S^1\). Moreover, \(\eta _N(S^1)=2\sin \frac{\pi }{2N}\).

In the theory of approximation and interpolation (for example, by splines or radial basis functions), the separation distance is often associated with some measure of “stability” of the approximation, while the mesh norm arises in the error of the approximation. In this context, the mesh-separation ratio  or simply mesh ratio3 
$$\begin{aligned} \gamma (\omega _N,A):=\eta (\omega _N, A)/\delta ^\rho (\omega _N), \end{aligned}$$
(3.2.5)
can be regarded as a “condition number” for \(\omega _N\) relative to A. We remark that if A is path connected, then for every \(N\ge 2\), we have \(\gamma (\omega _N, A)\ge 1/2\) provided that \(\omega _N\) consists of pairwise distinct points. This lower bound is attained, for example, if \(A=[0,1]\) and \(\omega _N=\omega _N^c\) as in Proposition 3.2.4.

Definition 3.2.5

If \(\{\omega _N\}_{N=2}^\infty \) is a sequence of N-point configurations such that \(\gamma (\omega _N, A)\) is bounded in N, then the sequence is said to be quasi-uniform on A .

Every infinite compact metric space has a quasi-uniform sequence of best-packing N-configurations as we now show.

Theorem 3.2.6

If \((A,\rho )\) is a compact infinite metric space, then for each \(N\ge 2\), there exists an N-point best-packing configuration \(\omega _N^*\) on A such that
$$\begin{aligned} \gamma (\omega _N^*, A)\le 1. \end{aligned}$$
(3.2.6)
In particular, this holds for any best-packing configuration \(\omega _N^*=\{x_1,\ldots , x_N\}\) having the minimal number of unordered pairs of points \(\{x_i, x_j\}\) such that \(\rho (x_i, x_j)=\delta ^\rho (\omega _N^*)\).

Proof

Let \(\omega _N^*\) be a best-packing N-point configuration on A having the minimal number of unordered pairs of points \(\{x_i, x_j\}\) such that \(\rho (x_i, x_j)=\delta ^\rho (\omega _N^*)\). Assume to the contrary that \(\eta (\omega _N^*, A)>\delta ^\rho (\omega _N^*)=\delta ^\rho _N(A)\). Select a point \(a\in A\) such that \(\rho (a, x_i)>\delta ^\rho _N(A)\) for \(i=1,\ldots , N\), and choose a point \(x_\ell \) from some pair \(\{x_k, x_\ell \}\) such that \(\rho (x_k, x_\ell )=\delta ^\rho _N(A)\). Let \(\omega _N'\) be the configuration obtained by replacing \(x_\ell \) in \(\omega _N^*\) by a. Notice that at least one pair \(\{x_i, x_j\}\) of points from \(\omega _N^*\) such that \(\rho (x_i, x_j)=\delta ^\rho _N(A)\) does not contain \(x_\ell \), since otherwise we would have \(\delta ^\rho (\omega _N')>\delta ^\rho (\omega _N^*)=\delta ^\rho _N(A)\). Hence, \(\omega _N'\) has at least one unordered pair of points \(\{x_i, x_j\}\) such that \(\rho (x_i, x_j)=\delta ^\rho _N(A)\) and furthermore, \(\delta ^\rho (\omega _N')=\delta ^\rho _N(A)\). However, \(\omega '_N\) has fewer such pairs than \(\omega _N^*\). This contradiction proves Theorem 3.2.6.    \(\square \)

Inequality (3.2.6) may fail for some A and some best-packing configurations. As we next show, there exist examples of compact metric spaces \((A, \rho )\) having non-quasi-uniform sequences of best-packing N-point configurations.

Example

Let A be the standard 1/3 Cantor set in [0,1] and let \(\rho \) be the Euclidean metric. For each \(N\in {\mathbb N}\), the set A is contained in the union of \(2^N\) disjoint intervals of length \(3^{-N}\) with endpoints
$$ 0=x_1^N<x^N_2<\ldots <x^N_{2^{N+1}}=1, $$
which belong to A. For any configuration of \(M_N:=2^N+1\) points in A, at least one of the intervals of length \(3^{-N}\) must contain at least two points from the configuration showing that \(\delta _{M_N}(A)\le 3^{-N}\). On the other hand, the configuration \(\omega _{M_N}:=\{x_1^N,\ldots , x_{M_N}^N=2/3\}\) is a best-packing configuration since \(\delta (\omega _{M_N})=\delta _{M_N}(A)= 3^{-N}\) and has mesh norm \(\eta (\omega _{M_N}, A)=1/3\). Thus
$$ \limsup _{N\rightarrow \infty }\ \sup \{\gamma (\omega _N, A)\mid \ \omega _N\subset A\ \mathrm{and}\ \delta (\omega _N)=\delta _N(A)\}=\infty . $$
In the general case one can easily verify that the sequence \(\{\delta ^\rho _N(A)\}_{N=2}^{\infty }\) is decreasing. Proposition 3.2.7 and Corollary 3.2.8 that we prove next immediately establish the connection between strict monotonicity of this sequence and the behavior of the mesh-separation ratio of best-packing sequences.

Proposition 3.2.7

Let \((A,\rho )\) be an infinite compact metric space and \(N\ge 2\) be an integer. There exists an N-point best-packing configuration \(\omega _N^*\subset A\) with \(\gamma (\omega _N^*, A)\ge 1\) if and only if \(\delta ^\rho _N(A)=\delta ^\rho _{N+1}(A)\).

Proof

Assume that \(\omega _N^*\) is a best-packing N-point configuration on A with \(\gamma (\omega _N^*, A)\ge 1\). Let \(a\in A\) be a point such that \(\min \limits _{x\in \omega _N^*}\rho (a,x)=\eta (\omega _N^*, A)\ge \delta ^\rho (\omega _N^*).\) Denote \(\omega '_{N+1}:=\omega _N^*\cup \{a\}\). Then \(\delta ^\rho (\omega '_{N+1})\ge \delta ^\rho (\omega _N^*)=\delta ^\rho _N(A)\). Since we also have \(\delta ^\rho (\omega '_{N+1})\le \delta ^\rho _{N+1}(A)\le \delta ^\rho _N(A)\), the equality \(\delta ^\rho _N(A)=\delta ^\rho _{N+1}(A)\) holds true.

Assume now that \(\delta ^\rho _N(A)=\delta ^\rho _{N+1}(A)\) for some \(N\ge 2\). Let \(\omega _{N+1}^*\) be an \((N+1)\)-point best-packing configuration on A and let \(\omega _N\) be an N-point subset of \(\omega _{N+1}^*\) obtained by deleting some arbitrary point x. Since
$$ \delta ^\rho (\omega _N)\ge \delta ^\rho (\omega _{N+1}^*)=\delta ^\rho _{N+1}(A)=\delta ^\rho _N(A), $$
we have that \(\omega _N\) is a best-packing N-point configuration on A. Since
$$ \eta (\omega _N,A)\ge \min _{y\in \omega _N}\rho (x, y)\ge \delta ^\rho (\omega _{N+1}^*)=\delta ^\rho _N(A)=\delta ^\rho (\omega _N), $$
we have \(\gamma (\omega _N, A)\ge 1\).    \(\square \)

The equation \(\delta ^\rho _N(A)=\delta ^\rho _{N+1}(A)\) can occur, for example, when \(A=S^2\), \(\rho \) is the Euclidean distance, and \(N=5\) (see Theorem 3.3.2 and Corollary 3.3.3) or \(N=11\). In the case \(N=11\) the best-packing configuration on \(S^2\) is obtained from the icosahedron by removing one point (see references at the end of this chapter). The icosahedron is known to be optimal in the case \(N=12\), see Section  5.7. This implies that \(\delta _{11}(S^2)=\delta _{12}(S^2)\).

Corollary 3.2.8

Let \((A,\rho )\) be an infinite compact metric space. The sequence \(\{\delta ^\rho _N(A)\}_{N=2}^{\infty }\) is strictly decreasing if and only if \(\gamma (\omega _N^*, A)<1\) for every \(N\ge 2\) and every best-packing N-point configuration \(\omega _N^*\) on A.

The statement below concerns the mesh-separation ratio of best-packing configurations that arise as cluster configurations of sequences of s-energy minimizing configurations.

Theorem 3.2.9

For a fixed \(N\ge 2\), let \(\omega _N\) be a cluster configuration as \(s\rightarrow \infty \) of a family \(\{\omega _N^s\}_{s>0}\) of N-point s-energy minimizing configurations on an infinite compact metric space \((A,\rho )\) (in the N-multi-set topology). Then \(\gamma (\omega _N, A)\le 1\).

Remark 3.2.10

The upper bound for \(\gamma (\omega _N, A)\) in this theorem can be attained even for the case when A is a sphere and \(\rho \) is the Euclidean metric. For \(N=11\) on \(S^2\), equality follows from the uniqueness result for best-packing of Böröczky. For \(N=5\) on \(S^2\), it follows from Theorem A.12.1.

Proof of Theorem 3.2.9 Let \(N\ge 2\) be fixed and, for \(s>0\), let \(\omega _N^s\) be an N-point s-energy minimizing configuration on A. There exists a point \(x_s\in \omega _N^s\) such that
$$ \sum _{y\in \omega _N^s\setminus \{x_s\}}\rho (x_s, y)^{-s}\ge N^{-1}E_s(\omega _N^s)\ge N^{-1}\delta ^\rho (\omega _N^s)^{-s}. $$
Denote by a a point in A such that \(\rho (y,a)\ge \eta (\omega _N^s, A)\), for all \(y\in \omega _N^s\), and let \(\widetilde{\omega }_N^s:=\left( \omega _N^s\setminus \{x_s\}\right) \cup \{a\}\). If it were true that \( \eta (\omega _N^s, A)> N^{2/s}\delta ^\rho (\omega _N^s),\) then we would have
$$ \sum \limits _{y\in \omega _N^s\setminus \{x_s\}}\rho (a, y)^{-s}\le \frac{N-1}{\eta (\omega _N^s, A)^s}<N^{-1}\delta ^\rho (\omega _N^s)^{-s}\le \sum _{y\in \omega _N^s\setminus \{x_s\}}\rho (x_s, y)^{-s}. $$
But this leads to the inequality \(E_s(\widetilde{\omega }_N^s)< E_s(\omega _N^s)\), which is a contradiction. Hence,
$$\begin{aligned} \eta (\omega _N^s, A)\le N^{2/s}\delta ^\rho (\omega _N^s). \end{aligned}$$
(3.2.7)
If now \(\omega _N\) is a cluster configuration as \(s\rightarrow \infty \) of a family of s-energy minimizing N-point configurations \(\{\omega _N^s\}_{s>0}\) on A, letting \(s\rightarrow \infty \) in (3.2.7) and using Proposition 3.1.2, we obtain \(\eta (\omega _N, A)\le \delta ^\rho (\omega _N)=\delta ^\rho _N(A)\), which yields the assertion of Theorem 3.2.9.    \(\square \)

3.3 Packing and Covering on the Sphere: Some Basics

In this section we present several fundamental results on packing and covering in the case when A is the d-dimensional sphere \(S^d\).

3.3.1 Best-Packing on the Sphere

The Tammes Problem  requires finding configurations of N points on \(S^2\) with the largest minimal pairwise distance (separation). This is the best-packing problem with \(A=S^2\) and \(\rho \) being the Euclidean distance in \(\mathbb R^3\). Solutions to the Tammes problem are known for \(N=2,3,..., 14\) and for \(N=24\). In particular, it is not difficult to verify that the solution for \(N=2\) is given by two antipodal points, \(\delta _2(S^2)=2\); for \(N=3\), by the vertices of an equilateral triangle inscribed in a great circle, \(\delta _3(S^2)=\sqrt{3}\); and for \(N=4\), by the vertices of a regular tetrahedron, \(\delta _4(S^2)=\sqrt{8/3}\). These best-packing configurations are a special case of the following basic result.

Theorem 3.3.1

For the sphere \(S^d\), \(d\ge 2\), and \(2\le N\le d+2\), best-packing configurations are uniquely given by the vertices of regular \((N-1)\)-simplices inscribed in \(S^d\) with centers at the origin. Furthermore, \(\delta _N(S^d)=\sqrt{\frac{2N}{N-1}}\).

Proof

Proposition 3.1.2 and Theorem  2.4.1 with \(K({ x},{ y})=\left| { x}-{ y}\right| ^{-s}\) imply that vertices of any regular \((N-1)\)-simplex inscribed in \(S^d\) and centered at the origin form a best-packing configuration. To show that no other N-point configuration is best-packing, let \(\omega _N=\{x_1,\ldots , x_N\}\subset S^d\) be such that \(\delta (\omega _N)=\sqrt{\frac{2N}{N-1}}\). Then \(x_i\cdot x_j\le -\frac{1}{N-1}\) for all \(1\le i\ne j\le N\). Assume to the contrary that at least one of these inequalities is strict. Then \(\sum \limits _{i\ne j}x_i\cdot x_j<-N\) and we get a contradiction since \(\left( \sum _{i=1}^{N}x_i\right) ^2<0\). Consequently, \(x_i\cdot x_j=-\frac{1}{N-1}\), \(i\ne j\), and so \(\left| x_i-x_j\right| =\sqrt{\frac{2N}{N-1}}\), \(i\ne j\), and \(\sum _{i=1}^{N}x_i=\mathbf{0}\).    \(\square \)

The solution to the best-packing problem on \(S^2\) for \(N=5\) points is a consequence of the next result due to Tammes.

Theorem 3.3.2

For the sphere \(S^2\), one has \(\delta _5(S^2)=\sqrt{2}\). Moreover, a five-point configuration \(\omega _5\) is a best-packing configuration on \(S^2\) if and only if it consists of two antipodal points and three vertices of a triangle inscribed in the associated equator whose angles are at least \(\pi /4\).

Proof

First note that any 5-point configuration \(\omega _5\) of the form described in the theorem has the property that the angle between vectors drawn from the origin to any two different points of \(\omega _5\) is at least \(\pi /2\), with some angles being exactly \(\pi /2\). Thus \(\delta (\omega _5)=\sqrt{2}\).

Now we assume that \(\omega _5=\{{ x}_1,\ldots ,{ x}_5\}\subset S^2\) satisfies \(\delta (\omega _5)\ge \sqrt{2}\) and we show that \(\omega _5\) must be of the stated form. For this purpose, note that it suffices to show that \(\omega _5\) contains a pair of antipodal points, for then the remaining three points must lie on the equator4 and form a triangle with all the angles at least \(\pi /4\).

So suppose to the contrary that no antipodal pair belongs to \(\omega _5\). We can also assume that \({ x}_1\) is located at the south pole; i.e., \({ x}_1=(0,0,-1)\). If the four remaining points all lie in the closed southern hemisphere, then since \(\delta (\omega _5)\ge \sqrt{2}\), they all must lie on the equator and form the vertices of a square, which violates the assumption of no antipodal points. Hence, at least one point, say \({ x}_2\), must lie strictly above the equator, and the remaining points in the closed northern hemisphere \(N_1\). Regarding \({ x}_2\) as the north pole of a closed hemisphere \(N_2\) and letting \(S_2\) denote the corresponding closed southern hemisphere, it follows that \({ x}_3\), \({ x}_4\), and \({ x}_5\) must all lie in \(N_1\cap S_2\). Without loss of generality we assume this intersection is of the form \(T_1\cup T_2\), where
$$ T_i:=\{(x,y, z)\in S^2 : 0\le (-1)^i x\le 1,\ 0\le z\le \lambda y\},\ \ \ i=1,2, $$
for some \(\lambda >0\). Clearly, two of the points, say \({ x}_3\) and \({ x}_4\), must lie in the same spherical triangle \(T_i\), say \(T_1\). Since for any two points \({ x},{ y}\in T_i\), we have \(\left| { x}-{ y}\right| ^2=2-2{ x}\cdot { y}<2\) unless \({ x}\cdot { y}=0\), it follows that one of these points must be at \((-1,0,0)\) and the other on the circular arc \(\{(0,y, z)\in S^2 : 0\le z \le \lambda y\}\). But then considering the triangle \(T_2\), the point \({ x}_5\) can only be located at (1, 0, 0), which contradicts the antipodal assumption.    \(\square \)
As a consequence of Proposition 3.1.2, we know that every cluster point as \(s\rightarrow \infty \) of 5-point s-energy minimizing configurations on \(S^2\) is a 5-point best-packing configuration on \(S^2\). However, Theorem 3.3.2 shows that there are infinitely many nonisometric best-packing configurations of 5 points, each of which is a possible cluster point. But as we prove in the Appendix (see Theorem A.12.1), all cluster configurations are isometric to the unique configuration
$$\begin{aligned} Q=\mathrm{SBP}(\infty ):=\{\mathrm{e}_1,-\mathrm{e}_1,\mathrm{e}_2,\mathrm{e}_3,-\mathrm{e}_3\}, \end{aligned}$$
(3.3.1)
where \(\mathrm{e}_1=(1,0,0)\), \(\mathrm{e}_2=(0,1,0)\), and \(\mathrm{e}_3=(0,0,1)\). Observe that this configuration has the maximum number of common pairwise distances (eight) of length \(\sqrt{2}\) among all 5-point best-packings.

Theorem 3.3.2 can also be used to solve the best-packing problem on \(S^2\) for \(N=6\) points.

Corollary 3.3.3

For the sphere \(S^2\), one has \(\delta _6(S^2)=\sqrt{2}\). Moreover, a six-point configuration on \(S^2\) is a best-packing configuration if and only if it forms the set of vertices of a regular octahedron inscribed in \(S^2\).

Proof

Observe that \(\delta _6(S^2)\le \delta _5(S^2)=\sqrt{2}\), see Theorem 3.3.2. For the set \(\overline{\omega }_6\) of the vertices of a regular octahedron inscribed in \(S^2\), we have \(\delta (\overline{\omega }_6)=\sqrt{2}=\delta _6(S^2)\). If \(\omega _6=\{{ x}_1,\ldots ,{ x}_6\}\subset S^2\) is any configuration such that \(\delta (\omega _6)= \sqrt{2}\), then the subconfiguration \(\omega '_5:=\{{ x}_1,\ldots ,{ x}_5\}\) satisfies \(\delta (\omega '_5)\ge \sqrt{2}\). In view of Theorem 3.3.2, the set \(\omega '_5\) is a best-packing configuration for \(N=5\), and hence contains two antipodal points, say \({ x}_1\) and \({ x}_2\). Then the remaining four points in \(\omega _6\) must form the vertices of a square that lies in the plane passing through the origin and perpendicular to the vectors \({ x_1}\) and \({ x}_2\); i.e., form the set of vertices of a regular octahedron.    \(\square \)

We next present a sharp estimate obtained by Fejes Tóth for the packing problem on the sphere \(S^2\). Throughout the rest of this section we define
$$ \theta _N:=\frac{\pi N}{6(N-2)}. $$

Theorem 3.3.4

If \(\omega _N\), \(N\ge 4\), is an arbitrary N-point configuration on \(S^2\),
$$\begin{aligned} \delta (\omega _N)\le \sqrt{4-\csc ^2\theta _N}. \end{aligned}$$
(3.3.2)

Remark 3.3.5

When \(N=4\) inequality (3.3.2) becomes \(\delta (\omega _4)\le \sqrt{8/3}\) and follows from Theorem 3.3.1. It turns into equality when \(\omega _4\) consists of the vertices of a regular simplex inscribed in \(S^2\) centered at the origin. For \(N=5\), inequality (3.3.2) is not sharp and follows from Theorem 3.3.2: \(\delta (\omega _5)\le \delta _5(S^2)=\sqrt{2}<\sqrt{4-(\csc (5\pi /18))^2}\). When \(N=6\), inequality (3.3.2) becomes \(\delta (\omega _6)\le \sqrt{2}\), which follows from Corollary 3.3.3. Equality occurs in (3.3.2) when \(\omega _6\) consists of the vertices of a regular octahedron inscribed in \(S^2\).

When \(N=12\), inequality (3.3.2) is \(\delta (\omega _{12})\le \sqrt{4-(\csc (\pi /5))^2}\). The pairwise dot-products between distinct vectors in a regular icosahedron \(\overline{\omega }_{12}\) inscribed in \(S^2\) are known to be \(1/\sqrt{5},-1/\sqrt{5},\) and \(-1\). Consequenly, \(\delta (\overline{\omega }_{12})=\sqrt{2-2/\sqrt{5}}=\sqrt{4-(\csc (\pi /5))^2}\) and we have the following result.

Corollary 3.3.6

A configuration that consists of the vertices of a regular icosahedron inscribed in \(S^2\) is a best-packing twelve-point configuration on \(S^2\). Furthermore, \(\delta _{12}(S^2)=\sqrt{2-2/\sqrt{5}}\).

Other proofs of Corollary 3.3.6 are given in Chapter  5 (see Example  5.5.5 and Corollary  5.7.5).

In the proof of Theorem 3.3.4 and of Theorem 3.3.15 below we will use the following notation, definitions, and auxiliary statements. Let \({ x}_1,\ldots ,{ x}_N\) be the points in \(\omega _N\) and let \(P_i\) be the tangent plane to \(S^2\) at point \({ x}_i\), \(i=1,\ldots , N\). Denote by \(H_i\) the closed half-space determined by the plane \(P_i\) that contains \(S^2\) and let
$$ U:=\bigcap \limits _{i=1}^{N}H_i. $$
A set that can be represented as the intersection of finitely many closed half-spaces is called a convex polyhedron . A set that can be represented as a convex hull of finitely many points is called a convex polytope . It is known that any bounded convex polyhedron is a convex polytope. If U is bounded, let \(G_i=U\cap P_i\) be the ith face of U and let \(\nu _i\) be the number of sides of the face \(G_i\). Let \(K_1,\ldots , K_N\) be closed spherical caps centered at points \({ x}_1,\ldots ,{ x}_N\) respectively of the same angular radius, which we denote by \(\varphi \). Denote by \(C_i\) the radial projection of the cap \(K_i\) onto the plane \(P_i\).
Set \(V({ x}_i):=\{{ x}\in S^2 : \left| { x}-{ x}_i\right| \le \left| { x}-{ x}_j\right| ,\ 1\le j\le N\}\), \(i=1,\ldots , N\), the Voronoi cell on \(S^2\) of the point \({ x}_i\) relative to the set \(\omega _N\). It is not difficult to see that for every point \({ y}\in P_i\),
$$\begin{aligned} \frac{{ y}}{\left| { y}\right| }\in V({ x}_i)\ \ \ \mathrm{if\ and \ only \ if}\ \ \ { y}\in G_i=U\cap P_i. \end{aligned}$$
(3.3.3)

Lemma 3.3.7

If \(\omega _N\) is not contained in any closed hemisphere of \(S^2\), then U is bounded, and hence is a convex polytope circumscribed about \(S^2\).

Proof

We only need to show that the convex polyhedron U is bounded. Since \(\omega _N\) is not contained in any closed hemisphere, for every \({ x}\in S^2\), there is \({ x}_i\in \omega _N\) such that \({ x}\cdot { x}_i>0\). Then the continuous function \(g({ x}):=\max \{{ x}\cdot { x}_j : 1\le j\le N\}\) has a positive minimum value \(\mu \) on \(S^2\). Hence, for every \({ y}\in U\), there is a point \({ x}_k\in \omega _N\) such that \({ x}_k\cdot { y}/\left| { y}\right| \ge \mu \). Since \({ y}\in H_k\), we have \(({ y}-{ x}_k)\cdot { x}_k\le 0\). Consequently, \(\mu \left| { y}\right| \le { x}_k\cdot { y}\le { x}_k\cdot { x}_k=1\) and we obtain that \(\left| { y}\right| \le 1/\mu \); i.e. U is bounded.    \(\square \)

Lemma 3.3.8

Let U be a convex polytope circumscribed about \(S^2\) having N faces and k edges. Then \(k\le 3(N-2)\).

Proof

Denote by v the number of vertices of U. Let \(q_i\), \(i=1,\ldots , v\), be the number of edges stemming out of the i-th vertex of U. It is not difficult to see that \(q_i\ge 3\) for every i. Since every edge of U stems from exactly two vertices, we can write \(3v\le q_1+\ldots + q_v=2k\). By the Euler’s Theorem for convex polytopes (see Theorem A.7.4), we have \(N-k+v=2\). Then \(3v=6-3N+3k\le 2k\), which implies that \(k\le 3N-6\).    \(\square \)

Lemma 3.3.9

For every \(N\ge 7\), we have \(\delta (\omega _N)<\sqrt{2}\) for every N-point configuration \(\omega _N\) on \(S^2\).

Proof

By Corollary 3.3.3, a configuration \(\omega _6^*\) is a best-packing 6-point configuration on \(S^2\) if and only if it consists of the vertices of a regular octahedron. Furthermore, \(\delta (\omega _6^*)=\sqrt{2}\). Since \(\eta (\omega _6^*, S^2)<\sqrt{2}\), we have \(\gamma (\omega _6^*, S^2)<1\). Then by Proposition 3.2.7, we have \(\delta _N(S^2)\le \delta _7(S^2)<\delta _6(S^2)=\sqrt{2}\).    \(\square \)

Proof of Theorem 3.3.4 In view of the Remark 3.3.5 it remains to establish inequality (3.3.2) for \(N\ge 7\). Without loss of generality, we will assume that the configuration \(\omega _N:=\{{ x}_1,\ldots ,{ x}_N\}\) is not contained in any closed hemisphere. Indeed, if some closed hemisphere with pole \({ y}_1\) contains \(\omega _N\), we move the point \({ x}_1\) into the point \(-{ y}_1\) creating a configuration \(\omega _N^1\) such that \(\delta (\omega _N)\le \delta (\omega _N^1)\) (one needs to take into account that \(\delta (\omega _N)<\sqrt{2}\) by Lemma 3.3.9). If \(\omega _N^1\) is still contained in some hemisphere with pole \({ y}_2\), since \(\delta (\omega _N^1)<\sqrt{2}\) (by Lemma 3.3.9), moving the point \({ x}_2\in \omega _N^1\) into the point \(-{ y}_2\) we obtain a configuration \(\omega _N^2\) such that \(\delta (\omega _N^1)\le \delta (\omega _N^2)\). Continuing this process we will obtain a configuration \(\omega _N^i=\{-{ y}_1,\ldots ,-{ y}_i,{ x}_{i+1},\ldots ,{ x}_N\}\), where the angle between vectors \(-{ y}_k\) and \(-{ y}_j\) is at least \(\pi /2\) whenever \(k\ne j\). By Lemma 3.3.9, we must have \(i\le 6\); i.e., this process will terminate giving a configuration \(\omega _N^l\) not contained in any closed hemisphere and such that \(\delta (\omega _N)\le \delta (\omega _N^l)\). Proving inequality (3.3.2) for \(\omega _N^l\) will establish it for \(\omega _N\).

We will use the notation introduced after Corollary 3.3.6. By Lemma 3.3.7, U is bounded, and hence, all its faces are convex polygons. Choose \(\varphi \) to be the maximal angular radius such that the caps \(K_1,\ldots , K_N\) are pairwise non-overlapping. Fix \(1\le i\le N\). Observe that \(K_i\subset V({ x}_i)\). In view of (3.3.3), \(G_i\) is the radial projection of \(V({ x}_i)\) onto \(P_i\) and, hence, \(C_i\subset G_i\). Let \(\nu _i\) be the number of edges of \(G_i\). Let also \({ z}_j\), \(j=1,\ldots ,\nu _i\) be the point of intersection of the circumference of \(C_i\) and the line in \(P_i\) passing through \({ x}_i\) perpendicular to the j-th edge of \(G_i\). Let \(L_j\) denote the tangent line to the circle \(C_i\) at the point \({ z}_j\) and let \(M_i\) denote the convex polygon circumscribed about \(C_i\) that is bounded by the lines \(L_j\), \(j=1,\ldots ,\nu _i\). Clearly, \(C_i\subset M_i\subset G_i\) and a similar inclusion holds for their radial projections \(K_i\), \(M_i'\), and \(V({ x}_i)\) onto \(S^2\). We partition the polygon \(M_i\) into \(2\nu _i\) right triangles by joining the point \({ x}_i\) with every vertex of \(M_i\) and with every point \({ z}_j\). The radial projections of these triangles are right spherical triangles \(T_1^i,\ldots , T_{2\nu _i}^i\) (since the plane through the points 0, \({ x}_i\), and \({ z}_j\) is perpendicular to the plane containing the point 0 and the line \(L_j\)).

Let \(\beta _n^i\) be the angle at the vertex \({ x}_i\) of the spherical triangle \(T_n^i\) and \(\gamma \) be the angle at the vertex v of \(T_n^i\) obtained as a radial projection of the corresponding vertex of the polygon \(M_i\). Let \(\alpha \) be the angle between vectors from 0 to \({ x}_i\) and from 0 to v. Clearly, \(0<\beta _n^i,\alpha <\pi /2\) and \(\gamma >0\). By the Second Cosine Theorem of Spherical Geometry (see Equation A.7.3 in the Appendix) we have
$$ \cos (\pi /2)=-\cos \beta _n^i\cos \gamma +\sin \beta _n^i\sin \gamma \cos \alpha , $$
which implies that \(\cos \gamma >0\) and hence, \(\gamma <\pi /2\). If j is chosen so that the radial projection of \({ z}_j\) is a vertex of \(T_n^i\) at its right angle, then the angle between vectors from 0 to the points \({ x}_i\) and \({ z}_j\) equals \(\varphi \). Applying Proposition A.7.2 we obtain that the spherical area5 \(\mathcal {H}_2(T_n^i)=\beta _n^i-\arcsin (\cos \varphi \sin \beta _n^i)\). Then
$$\begin{aligned} \begin{aligned} 4\pi&=\mathcal {H}_2(S^2)=\sum \limits _{i=1}^{N}\mathcal {H}_2(V({ x}_i))\ge \sum \limits _{i=1}^{N}\mathcal {H}_2(M_i')=\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\mathcal {H}_2(T_n^i)\\&=\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\left( \beta _n^i-\arcsin (\cos \varphi \sin \beta _n^i)\right) \\&=2\pi N-\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\arcsin (\cos \varphi \sin \beta _n^i). \end{aligned} \end{aligned}$$
Observe that \(2\sum _{i=1}^{N}\nu _i=4k\), where k is the number of edges of U, and \(\sum _{i=1}^{N}\sum _{n=1}^{2\nu _i}\beta _n^i=2\pi N\). Using the fact that the function \(y(t)=\arcsin (\cos \varphi \sin t)\) is concave downward on \((0,\pi /2)\), we then obtain
$$ 2\pi (N-2)\le \sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\arcsin (\cos \varphi \sin \beta _n^i)\le 4k\arcsin \left( \cos \varphi \sin \frac{\pi N}{2k}\right) . $$
Consequently,
$$ \cos \varphi \ge \frac{\sin \frac{\pi (N-2)}{2k}}{\sin \frac{\pi N }{2k}}. $$
Since every face of U has at least 3 edges, we have \(N\le \frac{2}{3}k\). By Lemma 3.3.8, we have \(2k\le 6(N-2)\). Observe that the function \(h(t):=\frac{\sin at}{\sin bt}\), where \(0<a<b\), is increasing on the interval \(\left( 0,\pi /(2b)\right) \) (to show that \(h'(t)>0\) one needs to observe that the function \(q(x)=\frac{\tan x}{x}\) is increasing on \((0,\pi /2)\)). Using this fact with \(a=\pi (N-2)\), \(b=\pi N\), \(t_1=1/(2k)\), and \(t_2=1/(6(N-2))\) we obtain that
$$ \cos \varphi \ge h(t_1)\ge h(t_2)=\frac{\sin \frac{\pi }{6}}{\sin \frac{\pi N}{6(N-2)}}=\frac{1}{2} \csc \theta _N . $$
Consequently,
$$ \delta (\omega _N)=2\sin \varphi =\sqrt{4-4\cos ^2\varphi }\le \sqrt{4-\csc ^2 \theta _N}. $$
   \(\square \)
The result presented in Theorem 3.3.4 can be restated in terms of the packing density of spherical caps on \(S^2\). Recall that the set
$$\begin{aligned} C({ a},\varphi ):=\{{ x}\in S^d : { x}\cdot { a}\ge \cos \varphi \} \end{aligned}$$
(3.3.4)
is called a spherical cap  of (angular) radius \(\varphi \) centered at point a. Recall that the area of the spherical cap \(C({ a},\varphi )\) equals \(2\pi (1-\cos \varphi )=\pi r^2\), where r is the Euclidean distance from the center a of the spherical cap to its boundary. We will call the number r the (Euclidean) radius  of the spherical cap. If a configuration \(\omega _N\subset S^2\) has minimal pairwise separation \(\rho =\delta (\omega _N)\), then the common Euclidean radius r of pairwise disjoint spherical caps centered at points of \(\omega _N\) satisfies \(r^2\le 2-\sqrt{4-\rho ^2}\). Then Theorem 3.3.4 inplies that \(r^2\le 2-\csc \theta _N\) and we have the following result.

Corollary 3.3.10

If \(K_1,\ldots , K_N\) are pairwise non-overlapping spherical caps on \(S^2\) of equal radii, then
$$\begin{aligned} \frac{1}{4\pi }\sum \limits _{i=1}^{N}\mathcal H_2(K_i)\le \frac{N}{2} \left( 1-\frac{1}{2} \csc \theta _N\right) . \end{aligned}$$
(3.3.5)
In Section  5.7 we give a different proof of the fact that for \(N=12\), the vertices of an inscribed regular icosahedron are best-packing points on \(S^2\). Best-packing configurations on \(S^2\) are also known for \(7\le N\le 11\), \(N=13, 14\), and \(N=24\) (see the Notes and Historical References section for the corresponding references).
Fig. 3.2

A near best-packing of 200 identical spherical caps on \(S^2\)

In Section  5.9 we obtain best-packing N-point configurations on \(S^d\) for \(d+3\le N\le 2d+2\). Other known best-packing configurations on a multidimensional sphere are also obtained in Section  5.7.

As with the Thomson problem (Section  2.4), finding exact solutions for the Tammes problem is essentially a hopeless endeavor for large values of N. Thus, we must resort to computational methods for generating near best-packing configurations. For \(N=200\) on \(S^2\), such points are illustrated in Figure 3.2.

3.3.2 Best-Covering on the Sphere

The best-covering problem on the sphere requires finding the quantity \(\eta _N(S^d)\) and N-point configurations on \(S^d\) that attain the infimum on the right-hand side of (3.2.3). In the case \(d=1\), for any \(N\in \mathbb N\), the configuration of N equally spaced points is best-covering. On \(S^2\) the solution to the best-covering problem is known for \(N=1,2,\ldots , 8, 10, 12,\) and 14. On \(S^3\) the best-covering problem has been solved for \(N=1,2,\ldots , 6\), and 8. On \(S^d\), \(d\ge 4\), the solution is known for \(N=1,2,\ldots , d+3\) (see the last section of this chapter for references to all these results). For \(N=200\) on \(S^2\), such points are illustrated in Figure 3.3.

We start by characterizing best-covering configurations of up to \(d+1\) points on \(S^d\). Note that any 1-point configuration on \(S^d\) is optimal.

Proposition 3.3.11

If \(2\le N\le d+1\), \(d\in \mathbb N\), then \(\eta _N(S^d)=\sqrt{2}\) and an N-point configuration on \(S^d\) is best-covering if and only if it is not contained in an open half-space relative to any hyperplane passing through the origin.

Proof

Let \(\omega _N\) be an arbitrary N-point configuration on \(S^d\). Since \(N\le d+1\), there is a d-dimensional affine subspace of \(\mathbb R^{d+1}\) that contains \(\omega _N\). Then there is a unit vector u such that \({ u}\cdot { x}\le 0\), for every \({ x}\in \omega _N\). Consequently, \(\left| { u}-{ x}\right| =\sqrt{2-2{ u}\cdot { x}}\ge \sqrt{2}\) and hence \(\eta (\omega _N, S^d)\ge \sqrt{2}\).

If \(\omega _N\) is contained in an open half-space relative to some hyperplane passing through the origin, there is a unit vector v such that \({ v}\cdot { x}<0\) for every \({ x}\in \omega _N\). Consequently, \(\left| { v}-{ x}\right| >\sqrt{2}\), \({ x}\in \omega _N\), and we have \(\eta (\omega _N, S^d)>\sqrt{2}\).

If \(\omega _N\) is not contained in an open half-space relative to any hyperplane passing through the origin, then for any unit vector a, there is a point \({ x}\in \omega _N\) such that \({ x}\cdot { a}\ge 0\). Then \(\min \limits _{{ x}\in \omega _N}{\left| { a}-{ x}\right| }\le \sqrt{2}\). Consequently, \(\eta (\omega _N, S^d)\le \sqrt{2}\), and the optimality of \(\omega _N\) follows.    \(\square \)

Fig. 3.3

A near best-covering of \(S^2\) by 200 identical spherical caps

The next result (see Remark 3.3.5 for the case \(d=2\)) shows the optimality of the vertices of the regular \((d+1)\)-simplex inscribed in \(S^d\) (\(N=d+2\)).

Theorem 3.3.12

Let \(d\in \mathbb N\). Then \(\eta _{d+2}(S^d)=\!\sqrt{\frac{2d}{d+1}}\). Moreover, the only best-covering \((d+2)\)-point configurations on \(S^d\) are given by the vertices of regular \((d+1)\)-simplices inscribed in \(S^d\).

For the proof of this theorem, we first establish the following auxiliary statement regarding coverings by spherical caps.

Lemma 3.3.13

Closed spherical caps of angular radius \(\varphi <\pi /2\) cover \(S^d\) if and only if the convex hull of their centers in \(\mathbb R^{d+1}\) contains the ball \(B[{ 0},\cos \varphi ]\).

Proof

Let \(\mathcal S\) be a collection of spherical caps of angular radius \(\varphi <\pi /2\) and let \(P=P(\mathcal S)\) be the convex hull of the centers of the caps from \(\mathcal S\).

Assume first that \(\mathcal S\) covers \(S^d\). Since \(\varphi <\pi /2\), the origin 0 is contained in the interior of P. Let H be the d-dimensional affine subspace that contains a given arbitrary facet of P (chosen arbitrarily). Let spherical cap \(C({ y},\psi )\) be the intersection of \(S^d\) with the half-space relative to H that does not contain the origin. Thus, \(\psi <\pi /2\). Since some spherical cap from the collection \(\mathcal S\) must contain y, the angular distance between y and some vertex of P is at most \(\varphi \). Then \(\psi \le \varphi \). The distance from H to the origin equals \(\cos \psi \ge \cos \varphi \). Hence, the ball \(B[{ 0}, \cos \varphi ]\) is a subset of the closed half-space relative to H that contains the origin. Since P is an intersection of such half-spaces, we have \(B[{ 0},\cos \varphi ]\subset P\).

Assume now that \(B[{ 0},\cos \varphi ]\subset P(\mathcal S)\) for some collection \(\mathcal S\) of spherical caps of angular radius \(\varphi <\pi /2\). Assume to the contrary that there is a point \({ z}\in S^d\) such that the (closed) cap \(C({ z},\varphi )\) contains no vertices of P. Then for some \(\theta \in (\varphi ,\pi /2)\), the cap \(C({ z},\theta )\) still contains no vertices of P. Let L be the affine subspace of \(\mathbb R^{d+1}\) that contains the boundary of \(C({ z},\theta )\) relative to \(S^d\) and let V be the half-space relative to L that contains the origin 0. Then \(B[{ 0},\cos \varphi ]\subset P\subset V\). On the other hand, L is at a distance of \(\cos \theta <\cos \varphi \) from 0, which implies that V does not contain \(B[{ 0},\cos \varphi ]\). This contradiction shows that the caps from \(\mathcal S\) cover \(S^d\).    \(\square \)

Proof of Theorem 3.3.12 The case \(d=1\) of Theorem 3.3.12 follows from Proposition 3.2.4. Therefore we will assume that \(d\ge 2\). Let \(N=d+2\). For a given point configuration \(\omega _N\subset S^d\), let \(\varphi =\varphi (\omega _N)\) be the minimal angular radius such that closed spherical caps \(C({ x},\varphi )\), \({ x}\in \omega _N\), cover \(S^d\).

Let \(\overline{\omega }_N\) be a configuration consisting of the vertices of a regular \((d+1)\)-simplex \(S^*\) inscribed in \(S^d\). We have \(\varphi (\overline{\omega }_N)<\pi /2\). Since the distance from the origin to any facet of \(S^*\) is \(1/(d+1)\), by Lemma 3.3.13, \(\varphi (\overline{\omega }_N)=\arccos (1/(d+1))\). Then \(\eta (\overline{\omega }_N, S^d)=\sqrt{2-2\cos \varphi (\overline{\omega }_N)}=\sqrt{\frac{2d}{d+1}}\).

Let \(\omega ^*_N\) be a best-covering configuration of \(d+2\) points on \(S^d\) and let S be the simplex with vertices from \(\omega ^*_N\). Then \(\varphi (\omega _N^*)\le \varphi (\overline{\omega }_N)<\pi /2\). By Lemma 3.3.13, the simplex S contains the ball \(B:=B[{ 0},\cos \varphi (\omega ^*_N)]\).

Assume to the contrary that \(\omega ^*_N\) is not a set of vertices of a regular \((d+1)\)-simplex inscribed in \(S^d\). Then there are distinct points \({ u}, { v}, { w}_1\in \omega ^*_N\) such that \(\left| { w}_1-{ u}\right| \ne \left| { w}_1-{ v}\right| \). Then \({ w}_1\notin H\), where H is the hyperplane that bisects perpendicularly the segment with endpoints u and v. Let \({ w}_2\), ..., \({ w}_d\) be the remaining vertices of S. For every point \({ x}\in \mathbb R^{d+1}\), let \({ x}'\) be the point symmetric to x relative to H and let \(\widetilde{x}:=\frac{1}{2} ({ x}+{ x}')\).

Let \(S'\) be the simplex symmetric to S relative to the hyperplane H and \(\mathcal A\) be the set of all lines in \(\mathbb R^{d+1}\) that are perpendicular to H. Recall that the Steiner symmetrization  of the simplex S with respect to the hyperplane H is the set
$$ K:=\bigcup _{l\in \mathcal A}\left( \frac{1}{2} (S\cap l)+\frac{1}{2} (S'\cap l)\right) . $$
Since \(B\subset S\), we have \(B=B'\subset S'\), and, consequently, \(B\subset K\).
We next show that \(K\subset \widetilde{S}\), where \(\widetilde{S}\) is the simplex with vertices \({ u},{ v},\widetilde{ w}_1,\ldots ,\widetilde{ w}_d\). Indeed, every point \({ z}\in K\) can be written as \({ z}=\frac{1}{2} ({ x}+{ y})\), where \({ x}\in S\cap l\) and \({ y}\in S'\cap l\) and \(l\bot H\). Since \({ x}\in S\), it can be written as \({ x}=\alpha ({ u}-{ v})+\gamma _0 { v}+\sum \limits _{i=1}^{d}\gamma _i{{ w}_i}\), where \(0\le \alpha \le \gamma _0\), \(0\le \gamma _i\), \(i=1,\ldots , d\), and \(\sum _{i=0}^{d}\gamma _i =1\). Then \({ x}'=\alpha ({ v}-{ u})+\gamma _0{ u}+\sum \limits _{i=1}^{d}\gamma _i{{ w}'_i}\). Using that \({ y}-{ x}'=\beta ({ u}-{ v})\) for some \(\beta \in \mathbb R\), we get \( { y}={ y}-{ x}'+{ x}'=(\beta -\alpha +\gamma _0){ u}+(\alpha -\beta ){ v}+\sum \limits _{i=1}^{d}{\gamma _i{ w}'_i} \), and so
$$\begin{aligned} { z}=\frac{1}{2} ({ x}+{ y})=\frac{1}{2} (\beta +\gamma _0){ u}+\frac{1}{2} (\gamma _0-\beta ){ v}+\sum \limits _{i=1}^{d}\gamma _i\widetilde{ w}_i. \end{aligned}$$
(3.3.6)
Since \({ y}\in S'\), we have \(\beta -\alpha +\gamma _0\ge 0\) and \(\alpha -\beta \ge 0\). Thus \(\beta +\gamma _0\ge \alpha \ge 0\), \(\gamma _0\ge \alpha \ge \beta \), and, hence, in (3.3.6) all coefficients are nonnegative; i.e., \({ z}\in \widetilde{S}\).

Consequently, \(B\subset K\subset \widetilde{S}\). However, since \({ w}_1\ne { w}'_1\), the vertex \(\widetilde{ w}_1\) of the simplex \(\widetilde{S}\) is located strictly inside \(S^d\). After moving radially every vertex of \(\widetilde{S}\) located strictly inside \(S^d\) onto \(S^d\), the hyperplanes containing all facets of \(\widetilde{S}\) except possibly one will be at a distance strictly greater than \(\cos \varphi (\omega ^*_N)\) from the origin. By slightly perturbing the vertices of the new simplex \(\widetilde{S}\) one can obtain a simplex T inscribed into \(S^d\) that contains a ball centered at the origin of radius strictly greater than \(\cos \varphi (\omega ^*_N)\). By Lemma 3.3.13, spherical caps centered at the vertices of T of some angular radius \(\psi <\varphi (\omega ^*_N)\) will still cover \(S^d\), which contradicts the optimality of \(\omega ^*_N\).

Thus, a best-covering \((d+2)\)-configuration can only lie at the vertices of regular \((d+1)\)-simplex inscribed in \(S^d\). Since the covering radius of the set of vertices of any regular \((d+1)\)-simplex is the same, we have \(\eta _N(S^d)=\eta (\omega _N, S^d)\) if and only if the convex hull of an N-point configuration \(\omega _N\) is a regular \((d+1)\)-simplex inscribed in \(S^d\).    \(\square \)

As the reader may have noticed, the above proof for best-covering for \(d+2\) points on \(S^d\) is more involved than the one for best-packing of \(d+2\) points. Furthermore, while the solution to the best-packing problem on \(S^d\) is known for \(d+3\le N\le 2d+2\) points (see Section  5.9), the best-covering problem remains open for \(N\ge d+4\) except for the cases on \(S^2\) mentioned in the first paragraph of this subsection and for the case \(d=3\) and \(N=8\). When \(d=3\) and \(N=8\) the vertices of the cross-polytope provide a solution for the best-covering of \(S^3\) (see the references at the end of the chapter). We next state the solution for best-covering of \(S^d\) by \(d+3\) points; for the proof see the references at the end of the chapter.

Theorem 3.3.14

A configuration of \((d+3)\) points consisting of two mutually orthogonal regular simplices on \(S^d\) with centers of mass at the origin having \(\lfloor (d+3)/2\rfloor \) and \(\lceil (d+3)/2\rceil \) points is a best-covering for \(S^d\).

The case \(d=2\) of Theorem 3.3.14 asserts that for \(N=5\), the best-covering configuration on \(S^2\) consists of the vertices of the triangular bipyramid BP.

We next present an estimate for the mesh-norm of an arbitrary N-point configuration on \(S^2\); its proof is given below after Corollary 3.3.17.

Theorem 3.3.15

If \(\omega _N\), \(N\ge 4\), is an arbitrary N-point configuration on \(S^2\), then
$$\begin{aligned} \eta (\omega _N, S^2)\ge \sqrt{2-\left( 2/\sqrt{3}\right) \cot \theta _N}, \end{aligned}$$
(3.3.7)
where \(\theta _N={\pi N}/[{6(N-2)}]\).

Remark 3.3.16

In the case \(N=4\), inequality (3.3.7) becomes \(\eta (\omega _4,4)\ge \sqrt{4/3}\), with equality attained by the vertices of a regular simplex inscribed in \(S^2\). This also follows from Theorem 3.3.12 in the case \(d=2\).

Direct calculations show that Theorem 3.3.15 implies the solution to the optimal covering problem on \(S^2\) for \(N=6\) and \(N=12\) stated next.

Corollary 3.3.17

A configuration consisting of the vertices of a regular octahedron (cross-polytope) inscribed in \(S^2\) is an optimal covering configuration on \(S^2\) for \(N=6\). A configuration consisting of the vertices of a regular icosahedron inscribed in \(S^2\) is an optimal covering configuration on \(S^2\) for \(N=12\). Also, \(\eta _6(S^2)=\sqrt{2-2/\sqrt{3}}\), \(\eta _{12}(S^2)=\sqrt{2-(2/\sqrt{3})\cot (\pi /5)}\).

As mentioned above, best-covering configurations on \(S^2\) are also known for \(N= 7,8, 10\), and 14. In particular, for \(N=7\), the optimal covering configuration consists of a point in the North pole, a point in the South pole, and five points in the corresponding equator located at the vertices of a regular pentagon. In the cases \(N=10\) and \(N=14\) the optimal covering configurations can be partitioned into antipodal pairs with points in some antipodal pair having circles around them that contain all the remaining points.

Proof of Theorem 3.3.15 The proof of this theorem is similar to the one for Theorem 3.3.4. We will use the notation introduced between Corollary 3.3.6 and Lemma 3.3.7. If \(\omega _N\) is contained in a closed hemisphere with pole x, then \(\eta (\omega _N, S^2)\ge \mathrm{dist}(-{ x},\omega _N)\ge \sqrt{2}\) and (3.3.7) follows. Thus we can assume that \(\omega _N\) is not contained in any closed hemisphere. By Lemma 3.3.7, U is bounded and, hence, all its faces \(G_i\) are convex polygons. We choose \(\varphi \) to be the smallest anglular radius such that caps \(K_1,\ldots , K_N\) cover \(S^2\). Fix \(1\le i\le N\) and observe that \(V({ x}_i)\subset K_i\). In view of (3.3.3), the radial projection of \(V({ x}_i)\) onto \(P_i\) is \(G_i\) and we have \({ x}_i\in G_i\subset C_i\). Let \({ y}_j\), \(j=1,\ldots ,\nu _i\), be the points of intersection with the circumference of \(C_i\) of the rays from \({ x}_i\) that pass through the vertices of \(G_i\) and let \(L_i\) be the convex polygon with vertices \({ y}_1,\ldots ,{ y}_{\nu _i}\). Then \(L_i\) is inscribed into \(C_i\) and contains \(G_i\). We partition the polygon \(L_i\) into \(2\nu _i\) right triangles by joining the point \({ x}_i\) with each vertex of \(L_i\) and with the midpoints \({ w}_1,\ldots ,{ w}_{\nu _i}\) of the sides of \(L_i\). Observe that the plane through the points 0, \({ w}_j\), and \({ x}_i\) is perpendicular to the plane that contains 0 and the side of \(L_i\) with the midpoint \({ w}_j\). Hence, the radial projections of these right triangles onto \(S^2\) are right spherical triangles, which we denote by \(T_1^i,\ldots , T_{2\nu _i}^i\). Let \(\beta _n^i\) be the angle of \(T_n^i\) at the vertex \({ x}_i\) and let \(\gamma \) be the angle of \(T_n^i\) at the vertex v which is the radial projection of the corresponding vertex of \(L_i\) onto \(S^2\). Observe that the angle between vectors from 0 to \({ x}_i\) and from 0 to v is \(\varphi \). Clearly, \(0<\beta _n^i,\varphi <\pi /2\) and \(\gamma >0\). Applying the Second Cosine Theorem of Spherical Geometry (see equation (A.7.3)) to the triangle \(T_n^i\), we have
$$ \cos (\pi /2)=-\cos \beta _n^i\cos \gamma +\sin \beta _n^i\sin \gamma \cos \varphi , $$
which implies that \(\cos \gamma >0\); i.e., \(\gamma <\pi /2\). Then the area of \(T_n^i\) is (see Proposition A.7.2)
$$ \mathcal {H}_2(T_n^i)=\beta _n^i-\arctan (\cos \varphi \tan \beta _n^i). $$
Since \(V({ x}_i)\subset L_i'\), where \(L_i'\) is the radial projection of \(L_i\) onto \(S^2\), we have
$$\begin{aligned} \begin{aligned} 4\pi&=\mathcal {H}_2(S^2)=\sum \limits _{i=1}^{N}\mathcal {H}_2(V({ x}_i))\le \sum \limits _{i=1}^{N}\mathcal {H}_2(L_i')=\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\mathcal {H}_2(T_n^i)\\ =&\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\left( \beta _n^i-\arctan (\cos \varphi \tan \beta _n^i)\right) \\ =&2\pi N-\sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\arctan (\cos \varphi \tan \beta _n^i). \end{aligned} \end{aligned}$$
(3.3.8)
Observe that \(\sum _{i=1}^{N}\nu _i=2k\), where k is the number of edges of U and that \(\sum _{i=1}^{N}\sum _{n=1}^{2\nu _i}\beta _n^i=2\pi N\). Since the function \(g(t)=\arctan (\cos \varphi \tan t)\) is convex on \((0,\pi /2)\), from (3.3.8) we have
$$ 2\pi (N-2)\ge \sum \limits _{i=1}^{N}\sum \limits _{n=1}^{2\nu _i}\arctan (\cos \varphi \tan \beta _n^i)\ge 4k\arctan \left( \cos \varphi \tan \frac{\pi N}{2k}\right) . $$
Consequently,
$$ \cos \varphi \le \frac{\tan \frac{\pi (N-2)}{2k}}{\tan \frac{\pi N}{2k}}. $$
Observe that the function \(h(t)=\frac{\tan at}{\tan bt}\), \(0<a<b\), is decreasing on the interval \((0,\pi /(2b))\) (to show that \(h'(t)<0\) one needs to use the fact that \(q(x)=\frac{\sin x}{x}\) is decreasing on \((0,\pi )\)). Since every face of U has at least three edges, we have \(N<k\). By Lemma 3.3.8, we have \(2k\le 6(N-2)\). Then with \(a=\pi (N-2)\), \(b=\pi N\), \(t_1=1/(2k)\) and \(t_2=1/(6(N-2))\) we obtain
$$ \cos \varphi \le h(t_1)\le h(t_2)=\frac{\tan \frac{\pi }{6}}{\tan \theta _N}=\frac{1}{\sqrt{3}}\cot \theta _N. $$
Since \(\varphi \) is the minimal angle such that caps \(K_1,\ldots , K_N\) cover \(S^2\), we have
$$ \eta (\omega _N, S^2)=\sqrt{2-2\cos \varphi }\ge \sqrt{2-\left( 2/\sqrt{3}\right) \cot \theta _N} $$
and (3.3.7) is proved.    \(\square \)

The result presented in Theorem 3.3.15 can be restated in terms of covering density of spherical caps of equal radii on \(S^2\). To do so we recall that the area of a spherical cap on \(S^2\) of Euclidean radius r equals \(\pi r^2\) (the area of a spherical cap on \(S^d\) can be obtained from formula ( 6.3.10) given further in the text using equation ( 6.3.3) or ( 6.3.1)).

Theorem 3.3.18

If \({K}_1,\ldots , K_N\subset S^2\), \(N\ge 4\), is a collection of spherical caps of equal radii that covers \(S^2\), then
$$\begin{aligned} \frac{1}{4\pi }\sum \limits _{i=1}^{N}\mathcal H_2(K_i)\ge \frac{N}{2}\left( 1-\frac{1}{\sqrt{3}}\cot \theta _N\right) , \end{aligned}$$
(3.3.9)
where \(\theta _N=\pi N/(6(N-2))\).

3.4 Packing and Covering in Euclidean Space

In this section we show that the problems of best-packing and best-covering in the plane are solved by the equi-triangular lattice  \(\varLambda ^*\) ; i.e., the additive subgroup of \(\mathbb R^2\) generated by vectors \({ v}_1=(1,0)\) and \({ v}_2=\left( \frac{1}{2},\frac{\sqrt{3}}{2}\right) \).
Fig. 3.4

An example of the Voronoi decomposition generated by a point configuration in the plane

Given a metric space \((A,\rho )\) and a discrete set \(\omega \subset A\), we set
$$ V_\rho ({ y}):=\{{ x}\in A : \rho ({ x},{ y} )\le \rho ({ x},{ y'}) , \, { y'}\in \omega \},\ \ \ { y}\in \omega . $$
The collection of sets \(\{V_\rho ({ x}) :{ x}\in \omega \}\) is called the Voronoi decomposition  of A with respect to \(\omega \) and \(\rho \) and we refer to a particular \(V_\rho ({ x})\) as the Voronoi cell  associated with the point \({ x}\in \omega \) (with respect to \(\omega \) and \(\rho \)), see Figure 3.4. The union of the Voronoi cells is A. When \(A=\mathbb R^p\) and \(\rho \) is the Euclidean distance, the Voronoi decomposition forms a partition of A; i.e., the interiors of the Voronoi cells are pairwise disjoint. This is not true in general. For example, if \(A=\mathbb R^2\), \(\rho ((x_1,y_1),(x_2,y_2))=\left| x_1-x_2\right| +\left| y_1-y_2\right| \), and \(\omega =\{(0,0),(1,1)\}\), then \(V((0,0))=\{y\le 0\}\cup \{x\le 0\}\cup \{x+y\le 1\}\) and \(V((1,1))=\{x\ge 1\}\cup \{y\ge 1\}\cup \{x+y\ge 1\}\) and, thus, the interiors of the Voronoi cells are not disjoint.
In the case of the Euclidean space \(\mathbb R^2\) and a finite configuration \(\omega \), the Voronoi cells are convex polygons or convex unbounded regions with a polygonal boundary. For example, the Voronoi decomposition of \(A=\mathbb R^2\) with respect to \(\varLambda ^*\) consists of congruent regular hexagons as shown in Figure 3.5.
Fig. 3.5

A fragment of the Voronoi decomposition formed by an equi-triangular lattice

The best-packing problem in the Euclidean space \(\mathbb R^p\) is formulated in the following way. Let \(\mathcal B\) be any packing in \(\mathbb R^p\); i.e., a collection of non-overlapping balls of the same radius in \(\mathbb R^p\). The (upper) density of the packing \(\mathcal B\)   is defined by
$$\begin{aligned} \varDelta (\mathcal B):=\limsup _{R\rightarrow \infty }{\frac{\sum _{B\in \mathcal B}\mathcal L_p\left( {B\cap [-R, R]^p}\right) }{(2R)^p}}. \end{aligned}$$
(3.4.1)
We remark that this notion of density is scale invariant; i.e., for any \(c>0\), \(\varDelta (\mathcal {B})=\varDelta (c\mathcal {B})\), where \(c\mathcal {B}:=\{c B:B\in \mathcal {B}\}\).
Roughly speaking, \(\varDelta (\mathcal B)\) represents the fraction of \(\mathbb R^p\) that is covered by \(\mathcal B\). The best-packing problem in \(\mathbb R^p\) is then to find the maximum possible packing density; that is, to find
$$\begin{aligned} \varDelta _p:=\sup \varDelta (\mathcal B),\ \ \ p\in \mathbb N, \end{aligned}$$
(3.4.2)
(where the supremum is taken over all packings \(\mathcal B\) in \(\mathbb R^p\)) as well as to determine packings \(\mathcal B\) that attain the least upper bound in (3.4.2).
Fig. 3.6

A fragment of an optimal packing of congruent disks in the plane

Clearly, \(\varDelta _1=1\). Regarding \(\varDelta _2\), Thue’s theorem of 1892 asserts that \(\varDelta _2={\pi }/{\sqrt{12}}\approx 0.90689968\) and that the packing of disks of radius 1/2 in \(\mathbb R^2\) centered at points of \(\varLambda ^*\) (see Figure 3.6) has the highest density. Completely rigorous proofs of this fact were published by Fejes Tóth in 1940, Segre and Mahler in 1944, and thereafter by several different authors (Davenport, Hsiang, and Hales). Following a plan suggested by Fejes Tóth and using a complex computer-aided proof, it was shown that \(\varDelta _3={\pi }/{\sqrt{18}}\approx 0.74048049\) (according to the Flyspeck Project team, the proof originally due to Hales and Ferguson and was completed in 1998). It has been fully validated by the Flyspeck team in 2014, who gave a formal proof based on the Isabelle and HOL Light formal proof assistants. This result implies that the face- centered cubic packing (FCC)  and the hexagonal close packing (HCP) (see Figure 3.7) are the densest sphere packings in \(\mathbb R^3\), resolving the more than 400-year-old Kepler conjecture.

Viazovska (in 2016) proved that \(\varDelta _8=\pi ^4/384\approx 0.25367\) and that the \(E_8\)-lattice sphere packing has the highest density.6 Cohn, Kumar, Miller, Radchenko, and Viazovska proved that \(\varDelta _{24}=\pi ^{12}/(12!)\approx 0.001930\) and that the Leech lattice sphere packing has the highest density.7 The quantity \(\varDelta _p\) is unknown for other values of p.
Fig. 3.7

Face centered cubic (left) and hexagonal close packing (right) in \(\mathbb R^3\)

The best (thinnest) covering problem in the Euclidean space \(\mathbb R^p\) is formulated in the following way. Let \(\mathcal C\) be any covering of \(\mathbb R^p\); i.e., a collection of closed balls in \(\mathbb R^p\) of the same radius whose union equals \(\mathbb R^p\). The density of the covering \(\mathcal C\), \(\varDelta (\mathcal C)\), is defined by the limit (3.4.1) provided that the limit exists. Define
$$\begin{aligned} \varGamma _p:=\inf \varDelta (\mathcal C),\ \ \ p\in \mathbb N, \end{aligned}$$
(3.4.3)
where the infimum is taken over all coverings of \(\mathbb R^p\) whose density \(\varDelta (\mathcal C)\) exists. The best covering problem requires finding the minimal covering density (3.4.3) and the thinnest (or most economical) coverings; i.e., the ones that attain the greatest lower bound in (3.4.3).

Clearly \(\varGamma _1=1\). A result of Kershner implies that \(\varGamma _2=2\pi /\sqrt{27}\) and that the covering by closed disks of radius 1 centered at points of \(\sqrt{3}\varLambda ^*\) has the smallest density. The quantity \(\varGamma _p\) is unknown for \(p>2\).

To derive the highest sphere packing density for the plane we will need the following auxiliary statement.

Lemma 3.4.1

Let \(\varOmega \) be a convex polygon in \(\mathbb R^2\) with \(\kappa \) sides and \(\omega _N=\{{ x}_1,\ldots , { x}_N\}\) a set of N points in \(\varOmega \). Let \(\{V_1, \ldots , V_N\}\) denote the Voronoi decomposition of \(\varOmega \) with respect to \(\omega _N\) (where \(V_i\) is the Voronoi cell associated with \({ x}_i\)) and let \(\nu _i\) denote the number of sides of \(V_i\). Then
$$\begin{aligned} \sum _{i=1}^N\nu _i\le 6N+\kappa -6. \end{aligned}$$
(3.4.4)

Proof

Let E and V denote the number of edges and vertices in the planar graph (i.e., the Voronoi diagram ) associated with this Voronoi decomposition. Every vertex in the diagram that is not a vertex of \(\varOmega \) must be an endpoint of at least 3 edges and each edge contains exactly two vertices and so we obtain the following estimate:
$$\begin{aligned} 2E\ge 3(V-\kappa )+2\kappa =3V-\kappa . \end{aligned}$$
(3.4.5)
Euler’s characteristic formula for this planar graph (or see Theorem A.7.4) can be used to show that (here the number of faces is N)
$$\begin{aligned} N+V-E=1. \end{aligned}$$
(3.4.6)
Combining (3.4.5) and (3.4.6) we obtain
$$ 1=N+V-E\le N+(2E+\kappa )/3-E=N-(1/3)E+\kappa /3 $$
or, equivalently,
$$\begin{aligned} 2E\le 6N+2\kappa -6. \end{aligned}$$
(3.4.7)
Noting that every edge not on the boundary of \(\varOmega \) is contained in two Voronoi cells while those on the boundary are contained in one cell completes the proof:
$$\sum _{i=1}^N\nu _i=2E-\#\{\text {edges contained in boundary of } \varOmega \} $$
$$ \le 2E-\kappa \le 6N+\kappa -6.$$
   \(\square \)

The next result states the “obvious” fact that a regular n-gon has the minimum (maximum) area among all n-gons (i.e., polygons with n sides) with the same in-radius (circumradius). We provide a proof for the convenience of the reader.

Lemma 3.4.2

Suppose \(\varOmega \) is a convex polygon in \(\mathbb R^2\) with n sides.

  1. (a)

    If \(\varOmega \) contains a unit-radius disk, then \(\mathcal L_2(\varOmega )\ge a_I(n):=n\tan (\pi /n)\).

     
  2. (b)
    If \(\varOmega \) is contained in a unit-radius disk, then
    $$ \mathcal L_2(\varOmega )\le a_C(n):=(n/2)\sin (2\pi /n). $$
     

The reader may verify that \(a_I(n)\) is the area of a regular n-gon with in-radius 1 and that \(a_C(n)\) is the area of a regular n-gon with circumradius equal to 1.

Proof

Let \(\varOmega \) be a convex n-gon with vertices \(A_1,\ldots , A_{n}\) arranged in a counterclockwise ordering (we also assume \(A_{n+1}=A_1\)). Suppose \(\varOmega \) contains a unit-radius disk U with center O. If some of the sides of \(\varOmega \) do not intersect U, then we may find another n-gon that is a proper subset of \(\varOmega \) and contains U. Hence, we may assume, without loss of generality, that each edge \(A_{j}A_{j+1}\) is tangent to U at some point \(P_j\). Let \(\alpha _{j, 0}\) denote the (acute) angle \(\angle A_jOP_{j}\) and let \(\alpha _{j, 1}\) denote the angle \(\angle P_j O A_{j+1}\) for \(j=1,\ldots , n\). Then the area of the triangle \(\triangle OA_jA_{j+1}\) equals \((1/2)\left( \tan \alpha _{j, 0}+\tan \alpha _{j, 1}\right) \) and so
$$ {\mathcal L_2}(\varOmega )=\frac{1}{2}\sum _{j=1}^n\sum _{l=0,1} \tan \alpha _{j, l}\ge n \tan (\pi /n), $$
where the last inequality follows from the convexity of the tangent function on \([0,\pi /2)\) and the observation that \(\sum _{j=1}^n\sum _{l=0,1} \alpha _{j, l}=2\pi \).
Similarly, if \(\varOmega \) is contained in a unit-radius disk, we may assume without loss of generality that the vertices \(A_1, \ldots , A_{n}\) lie on the boundary of U. Letting \(\beta _j\) denote the angle \(\angle A_jOA_{j+1}\) for \(j=1,\ldots , n\), we have
$$ {\mathcal L_2}(\varOmega )=\frac{1}{2}\sum _{i=1}^n\sin \beta _j \le (n/2) \sin (2\pi /n), $$
where the last inequality follows from the concavity of the sine function on \([0,\pi ]\) and the fact that \(\sum _{i=1}^n\beta _i=2\pi \).

   \(\square \)

We next prove sharp bounds for the best-packing and best-covering problems in the plane.

Lemma 3.4.3

If \(\varOmega \) is a convex polygon in \(\mathbb R^2\) with six or fewer sides and contains N pairwise disjoint open disks of radius \(r>0\), then
$$\begin{aligned} N\le \frac{\mathcal L_2(\varOmega )}{r^2\sqrt{12}}.\end{aligned}$$
(3.4.8)

Proof

Let \(\{B({ x}_i , r )\}_{i=1}^{N}\) be a collection of N pairwise disjoint open disks in \(\varOmega \) and let \(\{V_1,V_2,\ldots , V_N\}\) denote the collection of Voronoi cells in \(\varOmega \) associated with the centers \({ x}_i\), \(i=1,\ldots , N\). Let \(\nu _i\) be the number of vertices of \(V_i\). Then each \(V_i\) is a \(\nu _i\)-gon containing a disk of radius r and, in view of Lemma 3.4.1, we have
$$\begin{aligned} \sum _{i=1}^{N}{\nu _i}\le 6N. \end{aligned}$$
(3.4.9)
Observing also that the function \(a_I(x)=x\tan \frac{\pi }{x}\) is convex and strictly decreasing on the interval \((2,\infty )\) and using Lemma 3.4.2, we obtain
$$\begin{aligned} \begin{aligned} \mathcal L_2(\varOmega )&=\sum _{i=1}^{N}{\mathcal L_2(V_i)}\ge \sum _{i=1}^{N}{r^2a_I(\nu _i)}\ge N r^2 a_I\left( \frac{1}{N}\sum _{i=1}^{N}{\nu _i}\right) \\&\ge N r^2 a_I(6)=Nr^2\sqrt{12}, \end{aligned} \end{aligned}$$
which implies (3.4.8).    \(\square \)

Lemma 3.4.4

If \(\varOmega \) is a convex polygon with six or fewer sides and \(\varOmega \) is covered by N closed disks of radius \(r>0\), then
$$\begin{aligned} N\ge \frac{2\mathcal L_2(\varOmega )}{r^2\sqrt{27}}. \end{aligned}$$
(3.4.10)

Proof

Suppose \(\{B[{ y}_i, r]\}_{i=1}^{N}\) is a collection of N closed disks that covers \(\varOmega \). Let \({ x}_i\) be the point in \(\varOmega \) closest to \({ y}_i\), \(i=1,\ldots , N\). Then the collection \(\{B[{ x}_i, r]\}_{i=1}^{N}\) still covers \(\varOmega \). Indeed, for every \({ z}\in \varOmega \), there is an index i such that \(\left| { z}-{ y}_i\right| \le r\). By the choice of \({ x}_i\) and by Proposition A.1.1, we have \(\left| { z}-{ x}_i\right| \le \left| { z}-{ y}_i\right| \le r\), which implies that \({ z}\in B[{ x}_i, r]\). Let \(\{V_1,V_2,\ldots , V_N\}\) denote the Voronoi decomposition of \(\varOmega \) associated with the centers \({ x}_i\), \(i=1,\ldots , N\), and let \(\nu _i\) be the number of vertices of \(V_i\). For a fixed \(i\in \{1,\ldots , N\}\), let x be an arbitrary point in \(V_i\). Since x must be in \(B[{ x}_j, r]\) for some j and since \(\left| { x}-{ x}_i\right| \le \left| { x}-{ x}_j\right| \) it follows that \(V_i\subset B[{ x}_i, r]\).

A straightforward differentiation shows that \(a_c(x):= (x/2)\sin \frac{2\pi }{x}\) is strictly concave and increasing on \((2,\infty )\), and so using Lemmas 3.4.1 and 3.4.2, we obtain
$$\begin{aligned} \begin{aligned} \mathcal L_2(\varOmega )&=\sum _{i=1}^{N}{\mathcal L_2(V_i)}\le \sum _{i=1}^{N}{r^2a_c(\nu _i)}\le N r^2 a_c\left( \frac{1}{N}\sum _{i=1}^{N}{\nu _i}\right) \\&\le N r^2 a_c(6)=Nr^2 \sqrt{27}/2, \end{aligned} \end{aligned}$$
which implies (3.4.10).    \(\square \)

Using Lemmas 3.4.3 and 3.4.4 we now solve the best-packing and best-covering problems in \(\mathbb R^2\). For best-packing, we prove the following.

Theorem 3.4.5

The packing \(\mathcal B^*\) of circles in \(\mathbb R^2\) of radius 1 centered at points of the equi-triangular lattice \(2\varLambda ^*\) is the densest among all packings of circles of radius 1; i.e., \(\varDelta (\mathcal B^*)=\varDelta _2\), and so \(\varDelta _2=\pi /\sqrt{12}\).

Proof

Let \(\mathcal B\) be any packing of unit-radius disks in \(\mathbb R^2\) whose density is well-defined. Then by Lemma 3.4.3,
$$ \varDelta (\mathcal B)= \lim \limits _{R\rightarrow \infty }{\frac{1}{(2R)^2}\mathcal L_2\left( \cup _{B\in \mathcal B}{B\cap [-R, R]^2}\right) } $$
$$ \le \limsup \limits _{R\rightarrow \infty }{\frac{\pi }{(2R)^2}\cdot \# \{B\in \mathcal B : B\subset \left[ -R-2,R+2\right] ^2}\} $$
$$ \le \lim \limits _{R\rightarrow \infty }{\frac{\pi (2(R+2))^2}{\sqrt{12}(2R)^2}}=\frac{\pi }{\sqrt{12}}. $$
Hence, \(\varDelta _2\le \pi /\sqrt{12}\). For the packing \(\mathcal B^*\), we have \(\varDelta (\mathcal B^*)=\pi /\mathcal L_2(S_0)\), where \(S_0\) is the regular hexagon circumscribed about a circle of radius 1. Since \(\mathcal L_2(S_0)=\sqrt{12}\), we have \(\varDelta (\mathcal B^*)=\pi /\sqrt{12}\le \varDelta _2\) and hence, \(\varDelta _2=\varDelta (\mathcal B^*)=\pi /\sqrt{12}\).    \(\square \)

Theorem 3.4.6

The covering \(\mathcal C^*\) by closed disks in \(\mathbb R^2\) of radius 1 centered at points of the equi-triangular lattice \(\sqrt{3}\varLambda ^*\) is the thinnest among all coverings by unit-radius disks; i.e., \(\varDelta (\mathcal C^*)=\varGamma _2\), and so \(\varGamma _2=2\pi /\sqrt{27}\).

Proof

Let \(\mathcal C\) be any covering of \(\mathbb R^2\) by closed unit-radius disks whose density \(\varDelta (\mathcal C)\) is well defined. For every \(R>2\), set
$$ M_R:=\{B\in \mathcal C : B\subset [-R, R]^2\}. $$
The collection of disks \(M_R\) covers the square \([-R+2,R-2]^2\), and so applying Lemma 3.4.4 we obtain
$$\begin{aligned} \begin{aligned} \varDelta (\mathcal C)&=\lim \limits _{R\rightarrow \infty }{\frac{1}{(2R)^2}}{\sum _{B\in \mathcal C}\mathcal L_2\left( {B\cap [-R, R]^2}\right) }\\&\ge \limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^2}\sum \limits _{B\in M_R}\mathcal L_2(B)\\&\ge \limsup \limits _{R\rightarrow \infty }{\frac{\pi }{(2R)^2}}\cdot \#M_R\\&\ge \lim \limits _{R\rightarrow \infty }{\frac{2\pi (2R-4)^2}{(2R)^2\sqrt{27}}}=\frac{2\pi }{\sqrt{27}}. \end{aligned} \end{aligned}$$
Hence, \(\varGamma _2\ge 2\pi /\sqrt{27}\). For the covering \(\mathcal C^*\), we have \(\varDelta (\mathcal C^*)=\pi /\mathcal L_2(S_1)\), where \(S_1\) is the regular hexagon inscribed in the circle of radius 1. Since \(\mathcal L_2(S_1)=\sqrt{27}/2\), we have \(\varDelta (\mathcal C^*)=2\pi /\sqrt{27}\le \varGamma _2\). Consequently, \(\varGamma _2=\varDelta (\mathcal C^*)=2\pi /\sqrt{27}\).    \(\square \)
Optimality of the hexagonal pattern arises in nature as illustrated in Figure 3.8 showing ancient mathematicians at work.
Fig. 3.8

Ancient mathematicians at work

3.5 Mesh Ratio Minimization for Configurations in Euclidean Space

In this section, we introduce one more discrete geometric problem related to both packing and covering. For an arbitrary point configuration \(X\subset \mathbb R^p\) such that \(\delta (X)=\inf \{\left| x-y\right| : x, y\in X,\ x\ne y\}>0\), we consider its mesh ratio (as defined in (3.2.5)) \(\gamma (X,\mathbb R^p)=\eta (X,\mathbb R^p)/\delta (X)\), where \(\eta (X,\mathbb R^p)\) is the covering radius of the set X with respect to \(\mathbb R^p\), see (3.2.1). It is required to find the quantity
$$\begin{aligned} \varTheta _p:=\inf \{\gamma (X,\mathbb R^p):\, X\subset \mathbb R^p,\,\delta (X)>0\} \end{aligned}$$
(3.5.1)
and point sets \(X^*\) that attain the infimum on the right-hand side of (3.5.1).

It is not difficult to see that \(\varTheta _1=1/2\). In general, the quantity in (3.5.1) can be estimated via the highest sphere packing density and the minimal covering density of \(\mathbb R^p\) in the following way.

Theorem 3.5.1

For \(p\in \mathbb N\),
$$\begin{aligned} \varTheta _p\ge \frac{1}{2}\left( \frac{\varGamma _p}{\varDelta _p}\right) ^{1/p}. \end{aligned}$$
(3.5.2)
If there exists a set \(X\subset \mathbb R^p\) that is comprised of centers of balls that form a densest packing as well as balls that form a thinnest covering of \(\mathbb R^p\), then equality holds in (3.5.2) and X attains the infimum in (3.5.1).

Proof

Let \(X\subset \mathbb R^p\) be an arbitrary set with \(\delta (X)>0\). Setting \(r:=\delta (X)/2\), assume first that \(b:=\eta (X,\mathbb R^p)<\infty \). Then the collection of balls \(\{B[x, r]\}_{x\in X}\) is a packing in \(\mathbb R^p,\) while the collection of balls \(\{B[x, b]\}_{x\in X}\) is a covering of \(\mathbb R^p\). With these facts in mind and setting \(C_R:=[-R, R]^p\), we have
$$\begin{aligned} \begin{aligned} \varGamma _p&\le \limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{x\in X}\mathcal L_p(B[x, b]\cap C_R)\\&=\limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{x\in X\cap C_R}\mathcal L_p(B[x, b])\\&=\left( \frac{b}{r}\right) ^p\limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{x\in X\cap C_R}\mathcal L_p(B[x, r])\\&=\left( \frac{b}{r}\right) ^p\limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{x\in X}\mathcal L_p(B[x, r]\cap C_R)\le \left( \frac{b}{r}\right) ^p\varDelta _p, \end{aligned} \end{aligned}$$
(3.5.3)
where the first and the third equalities in (3.5.3) hold since the number of non-overlapping balls of radius b and radius r contained in the frames \(C_{R+b}\setminus C_{R-2b}\) and \(C_{R+r}\setminus C_{R-2r}\), respectively, becomes negligible compared to \(R^p\) as R gets large. Consequently,
$$\begin{aligned} \gamma (X,\mathbb R^p)=\frac{b}{2r}\ge \frac{1}{2}\left( \frac{\varGamma _p}{\varDelta _p}\right) ^{1/p}. \end{aligned}$$
(3.5.4)
Observe that in the case \(\eta (X,\mathbb R^p)=\infty \) inequality (3.5.4) holds trivially. Thus, in view of arbitrariness of the set X in (3.5.4), we obtain inequality (3.5.2).

If now X is a collection of centers of balls of some densest packing in \(\mathbb R^p\) and of some thinnest covering of \(\mathbb R^p\), then equality holds throughout (3.5.3). Hence \(\gamma (X,\mathbb R^p)=\frac{1}{2}\left( \frac{\varGamma _p}{\varDelta _p}\right) ^{1/p}\le \varTheta _p\), and so X is optimal in (3.5.1) and equality holds in (3.5.2).    \(\square \)

By Theorems 3.4.5 and 3.4.6, the equi-triangular lattice \(\varLambda ^*\) is the collection of centers of disks (of appropriate radius) of a densest packing in the plane and of a thinnest covering of the plane. Thus Theorem 3.5.1 implies the following result.

Corollary 3.5.2

The equi-triangular lattice \(\varLambda ^*\) minimizes the mesh ratio in the plane. Furthermore, \(\varTheta _2=1/\sqrt{3}\).

The following basic statement establishes the existence of saturated sphere packings in any dimension; i.e., sphere packings to which one cannot add another sphere of the same radius without overlapping with other spheres.

Proposition 3.5.3

For every \(p\in \mathbb N\), we have \(\frac{1}{2}\le \varTheta _p<1\).

Proof

Since \(\varGamma _p\ge \varDelta _p\), the inequality \(\varTheta _p\ge 1/2\) follows from inequality (3.5.2). To show that \(\varTheta _p<1\), we will construct a point set \(X\subset \mathbb R^p\) such that \(\eta (X,\mathbb R^p)<1\) and \(\delta (X)=1\). We start with the set \(X_0:=\mathbb Z^p\) (observe that \(\delta (X_0)=1\)). If \(\eta (X_0,\mathbb R^p)<1\), we stop and let \(X=X_0\). If \(\eta (X_0,\mathbb R^p)\ge 1\), since \(X_0\) is a periodic set, there is a point \(y_1\in \mathbb R^p\) such that \(\mathrm{dist}(y_1,X_0)=\eta (X_0,\mathbb R^p)\). Then we form the set \(X_1:=X_0\cup \left( y_1+\mathbb Z^p\right) \) and observe that \(\delta (X_1)=1\). If \(\eta (X_1,\mathbb R^p)<1\), we stop and let \(X=X_1\). If \(\eta (X_1,\mathbb R^p)\ge 1\), we find a point \(y_2\in \mathbb R^p\) such that \(\mathrm{dist}(y_2,X_1)\ge 1\) and form the set \(X_2:=X_1\cup (y_2+\mathbb Z^p)\). Observe that \(\delta (X_2)=1\). If we reach the k-th iteration, we will have the set \(X_k\) periodic with respect to the lattice \(\mathbb Z^p\) such that \(\delta (X_k)=1\). Hence, the family of balls \(\{B[x, 1/2]\}_{x\in X_k}\) will form a packing in \(\mathbb R^p\) whose density is \((k+1)\beta _p/2^p\). Since the density cannot be greater than 1, for some k, the process will terminate and we will have \(\eta (X_k,\mathbb R^p)<1\) showing that \(\varTheta _p\le \gamma (X_k,\mathbb R^p)<1\).    \(\square \)

Inequality (3.5.2) and Proposition 3.5.3 imply the following lower bound for packings.

Corollary 3.5.4

For \(p\in \mathbb N\),
$$\varDelta _p>\frac{\varGamma _p}{2^p}\ge \frac{1}{2^p}.$$

3.6 Bounds for the Sphere Packing Density in Arbitrary Dimensions

In this section, we present classical results by Minkowski and Blichfeldt that provide a lower and an upper bound respectively for the highest sphere packing density \(\varDelta _p\) for arbitrary dimension \(p\ge 2\). The two bounds (see Theorems 3.6.1 and 3.6.10) imply that \(\varDelta _p\) decays exponentially in p. A number of improvements were obtained later for each of these bounds (see the table on Section 3.8 for references on upper bounds), however, the exact base for the exponential decay remains unknown.

A packing of balls of equal radii in \(\mathbb R^p\) whose centers form a lattice is called a lattice packing . Denote by
$$ \varDelta ^*_p:=\sup \{\varDelta (\mathcal B): \mathcal B\ \mathrm{is\ a\ lattice\ packing\ in}\ \mathbb R^p\}. $$
The problem of finding densest lattice sphere packings is of independent interest. We have \(\varDelta _p=\varDelta ^*_p\) at least for \(p=1,2,3,8\), and 24. It is known that the equality \(\varDelta _p=\varDelta ^*_p\) fails for some values of p. For known lower bounds for \(\varDelta _p^*\) for arbitrary p, see the table on Section 3.8.

We start with the lower bound for lattice packings due to Minkowski.

Theorem 3.6.1

For \(p\in \mathbb N\), \(p\ge 2\),
$$ \varDelta _p\ge \varDelta ^*_p\ge \frac{\zeta (p)}{2^{p-1}}. $$

The proof of this lower bound follows from the more general Theorem 3.6.8 whose verification requires several lemmas.

Definition 3.6.2

Let \(\mathbb P\) denote the set of all prime numbers, and put
$$ H_q:=\{(1,x_2,\ldots , x_p)\in \mathbb Z^p : 0\le x_i<q,\ 2\le i\le p\},\ \ q\in \mathbb P. $$
For every point \(a\in H_q\), we define
$$ D_{a, q}:=\{ua+qz : u\in \mathbb Z,\ 1\le u<q,\ z\in \mathbb Z^p\} $$
and let \( U_{a, q} \) be the lattice in \(\mathbb R^p\) generated by vectors \(a, qe_2,\ldots , qe_p\), where \(e_i\) is the i-th standard basis vector in \(\mathbb R^p\), \(2\le i\le p\).

Lemma 3.6.3

For any \(a, b\in H_q\) such that \(a\ne b\), we have \(D_{a,q}\cap D_{b, q}=\emptyset \).

Proof

Assume to the contrary that there is some \(x\in D_a\cap D_b\). Then \(x=u_1a+qz_1=u_2b+qz_2\), where \(u_1,u_2\in \mathbb Z\), \(1\le u_1,u_2<q\), and \(z_1,z_2\in \mathbb Z^p\). Then
$$\begin{aligned} u_1a-u_2b=q(z_2-z_1). \end{aligned}$$
(3.6.1)
The first coordinate of the vector \(u_1a-u_2b\), which is \(u_1-u_2\), is divisible by q. Since \(-q<u_1-u_2<q\), we must have \(u_1-u_2=0\); i.e., \(u_1=u_2\). Thus (3.6.1) becomes
$$ u_1(a-b)=q(z_2-z_1). $$
Denote \(a=(1,x_2,\ldots , x_p)\) and let \(b=(1,y_2,\ldots , y_p)\). The i-th coordinate of the point \(u_1(a-b)\), \(2\le i\le p\), is \(u_1(x_i-y_i)\). It must also be divisible by q. Since the prime number q is not a divisor of \(u_1\), it must be a divisor of \(x_i-y_i\). However, \(-q<x_i-y_i<q\), and so \(x_i-y_i=0\); i.e., \(x_i=y_i\), \(2\le i\le p\). Consequently, we have \(a=b\) contradicting the choice of a and b. Thus, \(D_{a,q}\cap D_{b, q}=\emptyset \).    \(\square \)

Lemma 3.6.4

For every \(a\in H_q\), we have \(U_{a,q}\subset q\mathbb Z^p \cup D_{a, q}\), where the union is disjoint.8

Proof

Since the first coordinatre of every point in \(D_{a, q}\) is not divisible by q, we have \(q\mathbb Z^p\cap D_{a, q}=\emptyset \). Let \(v\in U_{a, q}\) be arbitrary point. Then \( v=\alpha _1 a+\alpha _2 q e_2+\ldots + \alpha _p qe_p \) for some integers \(\alpha _1,\ldots ,\alpha _p\). If \(\alpha _1\) is divisible by q, then \(v\in q\mathbb Z^p\). If \(\alpha _1\) is not divisible by q, then there are integers r and \(\alpha \), where \(1\le r<q\), such that \(\alpha _1=\alpha q+r\). Then \( v=ra+q(\alpha a+\alpha _2 e_2+\ldots +\alpha _p e_p)\in D_{a, q}. \) Consequently, \(v\in q\mathbb Z^p\cup D_{a, q}\).    \(\square \)

Let L be the set of all nonzero points from \(\mathbb Z^p\) whose coordinates have their greatest common divisor equal to 1. Observe that \(\mathbb Z^p\setminus \{0\}=\cup _{m=1}^{\infty }mL\), where the union is disjoint.

Lemma 3.6.5

If \(g:\mathbb R^p\rightarrow [0,\infty )\), \(p\ge 2\), is a compactly supported, bounded, and Riemann integrable function, then
$$ \lim \limits _{t\rightarrow 0^+}t^p\sum \limits _{x\in L}g(tx)=\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy. $$

Proof

Let \(\mu \) denote the M\(\mathrm{\ddot{o}}\)bius function (see Section A.3 in the Appendix for the definition). For every \(x\ne 0\), the function \(\tau (n):=g(nx)\) vanishes for every \(n\in \mathbb N\) sufficiently large. Then by Theorem A.3.2,
$$\begin{aligned} \begin{aligned} \sum \limits _{r=1}^{\infty }\mu (r)\!\!\!\!\sum \limits _{x\in \mathbb Z^p\setminus \{0\}}\!\!\!\!g(rx)&=\sum \limits _{r=1}^{\infty }\mu (r)\sum \limits _{m=1}^{\infty } \sum \limits _{x\in L}g(rmx)\\&=\sum \limits _{x\in L}\sum \limits _{m=1}^{\infty }\sum \limits _{r=1}^{\infty }\mu (r)g(rmx) =\sum \limits _{x\in L}g(x). \end{aligned} \end{aligned}$$
Put \(j(t):=t^p\!\!\!\!\!\sum \limits _{x\in \mathbb Z^p\setminus \{0\}}\!\!\!\!g(tx)\). Then for any \(t>0\),
$$\begin{aligned} J(t):=\sum \limits _{r=1}^{\infty }\frac{\mu (r)}{r^p}j(rt)=t^p\sum \limits _{r=1}^{\infty }\mu (r)\!\!\!\!\sum \limits _{x\in \mathbb Z^p\setminus \{0\}}\!\!\!\!g(rtx)=t^p\sum \limits _{x\in L}g(tx). \end{aligned}$$
(3.6.2)
Let \(R>0\) be such that g vanishes outside the cube \(C_R:=(-R, R)^p\). Since g is Riemann integrable on \(C_R\),
$$ \lim \limits _{t\rightarrow 0^+}j(t)=I:=\int _{C_R}\!\!g(y)\ \! dy=\int _{\mathbb R^p}\!\!g(y)\ \! dy. $$
Observe that \(j(t)=0\) for any \(t>R\). Let C be an upper bound for g on \(\mathbb R^p\). Then for \(t\in (0,R]\), we have
$$\begin{aligned} \begin{aligned} j(t)&\le \!\!\!\!\!\sum \limits _{x\in \mathbb Z^p\cap C_{R/t}}\!\!\!\!\!\!\!t^p g(tx)\le Ct^p\left( 2\lfloor R/t\rfloor +1\right) ^p\\&\le C(2R+t)^p\le C(3R)^p. \end{aligned} \end{aligned}$$
Choose an arbitrary \(\epsilon >0\). There are \(n_\epsilon \in \mathbb N\) and \(t_\epsilon >0\) such that
$$ \sum \limits _{r=n_\epsilon +1}^{\infty }\frac{\left| \mu (r)\right| }{r^p}<\epsilon \ \ \ \ \mathrm{and}\ \ \ \ \left| j(t)-I\right|<\epsilon ,\ \ 0<t<t_\epsilon , $$
and so for every \(0<t<t_\epsilon /n_\epsilon \), taking into account Theorem A.3.4, we obtain
$$\begin{aligned} \begin{aligned} \left| J(t)-\frac{I}{\zeta (p)}\right|&=\left| \sum \limits _{r=1}^{\infty }\frac{\mu (r)}{r^p}(j(rt)-I)\right| \\&\le \sum \limits _{r=1}^{n_\epsilon }\frac{\left| \mu (r)\right| }{r^p}\left| j(rt)-I\right| +\!\!\sum \limits _{r=n_\epsilon +1}^{\infty }\!\!\frac{\left| \mu (r)\right| }{r^p}\left( \left| j(rt)\right| +I\right) \\&\le \epsilon (\zeta (p)+C(3R)^p+I). \end{aligned} \end{aligned}$$
Consequently, \(\lim \limits _{t\rightarrow 0^+}J(t)=I/\zeta (p)\). In view of equality (3.6.2), we deduce the assertion of the lemma.    \(\square \)

Lemma 3.6.5 implies the following result.

Theorem 3.6.6

Let \(p\in \mathbb N\), \(p\ge 2\), and let \(g:\mathbb R^p\rightarrow [0,\infty )\) be a compactly supported, bounded, and Riemann integrable function. Then for every \(\epsilon >0\), there are a number \(q\in \mathbb P\) and a point \(a\in H_q\) such that the support of g contains no point from \(q^{\frac{1}{p}}\mathbb Z^p\) except possibly the origin and
$$ \sum \limits _{x\in U_{a, q}\cap L}g\left( q^{\frac{1}{p}-1}x\right) <\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy +\epsilon . $$

Proof

By Lemma 3.6.5, there is \(q_\epsilon \in \mathbb P\) such that for any \(q\in \mathbb P\), \(q>q_\epsilon \),
$$ \frac{1}{q^{p-1}}\sum \limits _{x\in L}g\left( q^{\frac{1}{p}-1}x\right) <\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy +\epsilon . $$
Choose any \(q\in \mathbb P\), \(q>q_\epsilon \). Taking into account Lemma 3.6.3, we have
$$ \frac{1}{q^{p-1}}\sum \limits _{a\in H_q}\sum \limits _{x\in L\cap D_{a, q}}g\left( q^{\frac{1}{p}-1}x\right) \le \frac{1}{q^{p-1}}\sum \limits _{x\in L}g\left( q^{\frac{1}{p}-1}x\right) <\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy +\epsilon . $$
Since \(\# H_q=q^{p-1}\), by the properties of the average, there is \(a=a_q\in H_q\) such that
$$ \sum \limits _{x\in L\cap D_{a, q}}g\left( q^{\frac{1}{p}-1}x\right) <\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy +\epsilon . $$
Using that \(\mathrm{supp}(g)\) is bounded, we can choose \(q\in \mathbb P\), \(q>q_\epsilon \), large enough so that \(\mathrm{supp}(g)\cap q^{\frac{1}{p}}\mathbb Z^p\subset \{0\}\). For such q,
$$ \sum \limits _{x\in L\cap q\mathbb Z^p}g\left( q^{\frac{1}{p}-1}x\right) =0. $$
Appealing to Lemma 3.6.4, we now have
$$\begin{aligned} \begin{aligned} \sum \limits _{x\in L\cap U_{a, q}}g\left( q^{\frac{1}{p}-1}x\right)&\le \sum \limits _{x\in L\cap q\mathbb Z^p}g\left( q^{\frac{1}{p}-1}x\right) +\sum \limits _{x\in L\cap D_{a, q}}g\left( q^{\frac{1}{p}-1}x\right) \\&=\sum \limits _{x\in L\cap D_{a, q}}g\left( q^{\frac{1}{p}-1}x\right) <\frac{1}{\zeta (p)}\int _{\mathbb R^p}g(y)\ \! dy +\epsilon . \end{aligned} \end{aligned}$$
   \(\square \)

Before stating the fundamental result on packing known as the Minkowski-Hlawka theorem we need the following definition.

Definition 3.6.7

We say that a set \(A\subset \mathbb R^p\) is a star-like domain if there is a point \(x_0\) in the interior of A (called a center of A) such that for every point \(x\in A\), the segment \([x_0,x]\) is contained in A and the segment \([x_0,x)\) is contained in the interior of A.

Theorem 3.6.8

Let \(p\in \mathbb N\), \(p\ge 2\), and let \(\varOmega \) be a compact and centrally symmetric star-like domain such that \(\mathcal L_p(\varOmega )< 2\zeta (p)\). Then there exists a full-rank lattice \(U\subset \mathbb R^p\) with fundamental parallelepiped of Lebesgue measure 1 such that \(\varOmega \) contains no nonzero point from U.

In particular, any convex, centrally symmetric, and compact set \(\varOmega \) with non-empty interior such that \(\mathcal L_p(\varOmega )< 2\zeta (p)\) satisfies the assumptions of Theorem 3.6.8.

Remark 3.6.9

In Theorem 3.6.8 one can take \(U=q^{\frac{1}{p}-1}U_{a, q}\) for a certain prime number q and point \(a\in H_q\) (see Definition 3.6.2).

Proof of Theorem 3.6.8 We first show that the origin is a center of the star-like domain \(\varOmega \). By the central symmetry of \(\varOmega \), the origin belongs to \(\varOmega \). By definition, \(\varOmega \) has its center at some point \(x_0\). Let \(x\in \varOmega \) be arbitrary point and let \(b\in [0,1)\). Since \(-x_0\) is also a center, the point \(z:=\frac{2b}{1+b}x+\frac{1-b}{1+b}(-x_0)\) is contained in \(\varOmega \). Then the point
$$ bx=bx+\frac{1-b}{2}(-x_0)+\frac{1-b}{2}x_0=\frac{1+b}{2}z+\frac{1-b}{2}x_0 $$
belongs to the interior \( \varOmega ^\circ \). Thus, \([0,x)\subset \varOmega ^\circ \) for every \(x\in \varOmega \); i.e., the origin is a center of \(\varOmega \).

We next show that \(\mathcal L_p(\partial \varOmega )=0\). Indeed, since the origin is a center of the set \((1+\delta )\varOmega \), \(\delta >0\), every \(x\in \varOmega \) is contained in the interior of \((1+\delta )\varOmega \). Then \(\mathcal L_p( \varOmega ^\circ )\le \mathcal L_p(\varOmega )\le (1+\delta )^p\mathcal L_p( \varOmega ^\circ )\). Letting \(\delta \rightarrow 0\), we have \(\mathcal L_p(\varOmega )=\mathcal L_p(\varOmega ^\circ )\). Since \(\varOmega \) is closed, \(\mathcal L_p(\partial \varOmega )=0\).

Now let \(\epsilon :=1-\frac{\mathcal L_p(\varOmega )}{2\zeta (p)}\). Since \(\mathcal L_p(\partial \varOmega )=0\), the characteristic function \(\chi _\varOmega \) of the set \(\varOmega \) is Riemann integrable. By Theorem 3.6.6, there exist a number \(q\in \mathbb P\) and a point \(a\in H_q\) such that \(\varOmega \cap q^{\frac{1}{p}}\mathbb Z^p=\{0\}\) and
$$\begin{aligned} \sum \limits _{x\in U_{a, q}\cap L}\chi _\varOmega \left( q^{\frac{1}{p}-1}x\right)< \frac{1}{\zeta (p)}\int _{\mathbb R^p}\chi _\varOmega (x)\ \! dx +\epsilon =1+\frac{\mathcal L_p(\varOmega )}{2\zeta (p)}<2. \end{aligned}$$
(3.6.3)
We claim that
$$\begin{aligned} M_q:=\{x\in U_{a, q}\cap L : q^{\frac{1}{p}-1}x\in \varOmega \}=\emptyset . \end{aligned}$$
(3.6.4)
Indeed, if there was a point \(y\in M_q\), by the central symmetry of the lattice \(U_{a, q}\) and of the sets L and \(\varOmega \), we would have \(-y\in M_q\). Then we would deduce that \(\sum \limits _{x\in U_{a, q}\cap L}\chi _\varOmega \left( q^{\frac{1}{p}-1}x\right) \ge 2\), contradicting (3.6.3).
We will now show that \(U:=q^{\frac{1}{p}-1}U_{a, q}\) is the required lattice. Since the matrix \([q^{\frac{1}{p}-1}a, q^{\frac{1}{p}}e_2,\ldots , q^{\frac{1}{p}}e_p]\) with columns \(q^{\frac{1}{p}-1}a, q^{\frac{1}{p}}e_2,\ldots , q^{\frac{1}{p}}e_p\) is lower triangular, the volume of the fundamental parallelepiped of the lattice U equals
$$ \left| U\right| =\mathrm{det}[q^{\frac{1}{p}-1}a, q^{\frac{1}{p}}e_2,\ldots , q^{\frac{1}{p}}e_p]=q^{\frac{1}{p}-1}q^{\frac{1}{p}}\cdot \ldots \cdot q^{\frac{1}{p}}=1. $$
Assume to the contrary that there is a nonzero vector \(v\in \varOmega \cap U\). Then \(v=q^{\frac{1}{p}-1}x\) for some nonzero \(x\in U_{a, q}\). There exist integers \(\alpha _1,\ldots ,\alpha _p\) such that
$$ x=\alpha _1a+\alpha _2 q e_2+\ldots +\alpha _p q e_p=(\alpha _1,\alpha _1 x_2+\alpha _2 q,\ldots ,\alpha _1 x_p+\alpha _p q). $$
Since \(\varOmega \cap q^{\frac{1}{p}}\mathbb Z^p=\{0\}\), we have \(x\notin q\mathbb Z^p\). Then \(\alpha _1\) is not divisible by q. Let \(m\in \mathbb N\) be the greatest common divisor of the coordinates of x. If \(m=1\), then \(x\in L\) and hence \(x\in M_q\) contradicting (3.6.4). If \(m>1\), since \(\alpha _1\) and \(\alpha _1x_i+\alpha _i q\) are both divisible by m, the number \(\alpha _i q\) is divisible by m, \(i=2,\ldots , p\). The prime number q is not a divisor of m because \(x\notin q\mathbb Z^p\). Since m is a divisor of \(\alpha _i q\) it must be a divisor of \(\alpha _i\), \(i=2,\ldots , p\). Thus, \(\alpha _i=my_i\), \(i=1,\ldots , p\), for some integers \(y_1,\ldots , y_p\). Consequently, \(x=mx'\), where
$$ x'=(y_1,y_1x_2+y_2q,\ldots , y_1x_p+y_pq). $$
Observe that \(x'=y_1a+y_2qe_2+\ldots +y_pqe_p\in U_{a, q}\) and that the greatest common divisor of the coordinates of \(x'\) is 1. Then \(x'\in U_{a, q}\cap L\) and the point \(q^{\frac{1}{p}-1}x'=\frac{1}{m}q^{\frac{1}{p}-1}x=\frac{1}{m}v\) belongs to \(\varOmega \) because \(\varOmega \) is a star-like domain. Consequently \(x'\in M_q\), contradicting (3.6.4). This contradiction shows that the intersection \(\varOmega \cap U\) contains no nonzero points.    \(\square \)
Proof of Theorem 3.6.1 Let \(B_\epsilon \), \(0<\epsilon <1\), be a closed ball in \(\mathbb R^p\) centered at the origin of Lebesgue measure \(2\zeta (p)(1-\epsilon )\) and let \(r_\epsilon \) be its radius. By Theorem 3.6.8 there exists a full-rank lattice U in \(\mathbb R^p\) whose fundamental parallelepiped has Lebesgue measure 1 such that \(B_\epsilon \) contains no nonzero points from U. Then \(\delta (U)\ge r_\epsilon \). The collection of balls \(\{B[v, r_\epsilon /2]\}_{v\in U}\) forms a packing \(\mathcal B\) in \(\mathbb R^p\) with density
$$ \varDelta (\mathcal B)=\frac{\mathcal L_p(B[0,r_\epsilon /2])}{\left| U\right| }=\frac{\mathcal L_p(B_\epsilon )}{2^p}=\frac{\zeta (p)}{2^{p-1}}(1-\epsilon ). $$
On letting \(\epsilon \rightarrow 0\) the assertion of Theorem 3.6.1 follows.    \(\square \)

We conclude this section with the upper bound for \(\varDelta _p\) obtained by Blichfeldt.

Theorem 3.6.10

For \(p\in \mathbb N\),
$$ \varDelta _p\le \frac{p+2}{2^{1+p/2}}. $$

To prove this theorem we will need the following two auxiliary statements.

Lemma 3.6.11

If \(x_1,\ldots , x_k\) are arbitrary points in \(\mathbb R^p\) such that \(\left| x_i-x_j\right| \ge 2\), \(1\le i\ne j\le k\), then for any point \(a\in \mathbb R^p\),
$$ \sum \limits _{i=1}^{k}\left| x_i-a\right| ^2\ge 2(k-1). $$

Proof

Let \(y_i:=x_i-a\), \(i=1,\ldots , k\). Then for any \(i\ne j\), we have \(\left| y_i-y_j\right| ^2=\left| x_i-x_j\right| ^2\ge 4\). Adding these inequalities we obtain
$$\begin{aligned} \begin{aligned} \sum \limits _{i=1}^{k}\sum \limits _{j=1\atop j\ne i}^{k}\left| y_i-y_j\right| ^2&=\sum \limits _{i=1}^{k}\sum \limits _{j=1}^{k}\left( y_i^2-2y_i\cdot y_j+y_j^2\right) \\&=2k\sum \limits _{i=1}^{k}y_i^2-2\sum \limits _{i=1}^{k}\sum \limits _{j=1}^{k}y_i\cdot y_j\\&=2k\sum \limits _{i=1}^{k}y_i^2-2\left| \sum \limits _{i=1}^{k}y_i\right| ^2\ge 4k(k-1). \end{aligned} \end{aligned}$$
Thus \( \sum _{i=1}^{k}\left| x_i-a\right| ^2=\sum _{i=1}^{k}y_i^2\ge 2(k-1). \)    \(\square \)

Lemma 3.6.12

Let \(\mathcal B\) be any packing in \(\mathbb R^p\) of unit radius balls. If \(\mathcal B_R\), \(R>0\), denotes the collection of balls from \(\mathcal B\) that are contained in the cube \([-R, R]^p\), then
$$ \# \mathcal B_R< 2^{p/2-1}\frac{p+2}{ \beta _p}(R+1)^p. $$

Proof

Let
$$ \varphi (x):={\left\{ \begin{array}{ll} 2-\left| x\right| ^2, &{} \left| x\right| \le \sqrt{2},\\ 0, &{} \left| x\right| >\sqrt{2}.\\ \end{array}\right. } $$
Denote by \(c_B\) the center of a ball B from the packing \(\mathcal B\) and let
$$\begin{aligned} u(x):=\sum \limits _{B\in \mathcal B}\varphi (x-c_B). \end{aligned}$$
(3.6.5)
If \(x\in \mathbb R^p\) is an arbitrary point, the sum on the right-hand side of (3.6.5) contains finitely many nonzero terms; indeed, there are at most finitely many balls from the packing \(\mathcal B\) that are contained in the ball \(B[x,\sqrt{2}+1]\). Consequently, at most finitely many centers of balls from \(\mathcal B\) are at a distance of no more than \(\sqrt{2}\) from x. Denote them by \(b_1,\ldots , b_k\). Then
$$ u(x)=\sum \limits _{i=1}^{k}\varphi (x-b_i)=\sum \limits _{i=1}^{k}(2-\left| x-b_i\right| ^2)=2k-\sum \limits _{i=1}^{k}\left| x-b_i\right| ^2. $$
Since \(\left| b_i-b_j\right| \ge 2\), \(i\ne j\), by Lemma 3.6.11,
$$ u(x)\le 2k-2(k-1)=2, \ \ \ x\in \mathbb R^p. $$
Let \(\mathcal B_R\) be the collection of balls from the packing \(\mathcal B\) that are contained in the cube \([-R, R]^p\). Then balls \(B[c_B,\sqrt{2}]\), \(B\in \mathcal B_R\) are contained in the cube \(D_R:=[-R-\sqrt{2}+1,R+\sqrt{2}-1]^p\). Since \(\varphi \) is nonnegative, we have
$$\begin{aligned} \begin{aligned} 2\mathcal L_p(D_R)&\ge \int \limits _{D_R} u(x)\ \! dx \ge \int _{D_R}\left( \sum \limits _{B\in \mathcal B_R}\varphi (x-c_B)\right) \ \! dx\\&=\sum \limits _{B\in \mathcal B_R}\int \limits _{B[c_B,\sqrt{2}]}\varphi (x-c_B)\ \! dx=\#\mathcal B_R\!\! \int \limits _{B[0,\sqrt{2}]}\!\!\!\varphi (x)\ \! dx. \end{aligned} \end{aligned}$$
Since
$$ \int \limits _{B[0,\sqrt{2}]}\!\!\!\varphi (x)\ \! dx=p\beta _p \int _{0}^{\sqrt{2}}(2-t^2)t^{p-1}\ \! dt=2^{2+p/2}\frac{\beta _p}{p+2}, $$
we have
$$ \# \mathcal B_R\le \frac{(p+2)\mathcal L_p(D_R)}{2^{1+p/2}\beta _p}<2^{p/2-1}\frac{p+2}{\beta _p}(R+1)^p. $$
   \(\square \)
Proof of Theorem 3.6.10 Let \(\mathcal B\) be an arbitrary packing of unit radius balls in \(\mathbb R^p\) whose density \(\varDelta (\mathcal B)\) exists. Recall that \(\mathcal B_{R+2}\) is the collection of balls in \(\mathcal B\) that are contained in the cube \([-R-2,R+2]^p\). Taking into account Lemma 3.6.12, we have
$$\begin{aligned} \begin{aligned} \varDelta (\mathcal B)&=\lim \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{B\in \mathcal B}\mathcal L_p(B\cap [-R, R]^p)\le \limsup \limits _{R\rightarrow \infty }\frac{1}{(2R)^p}\sum \limits _{B\in \mathcal B_{R+2}}\mathcal L_p(B)\\&=\beta _p\limsup \limits _{R\rightarrow \infty }\frac{\# \mathcal B_{R+2}}{(2R)^p}\le 2^{p/2-1}(p+2)\limsup \limits _{R\rightarrow \infty }\frac{(R+3)^p}{(2R)^p}=\frac{p+2}{2^{1+p/2}}. \end{aligned} \end{aligned}$$
In view of arbitrariness of the packing \(\mathcal B\), we deduce Theorem 3.6.10.    \(\square \)

3.7 Asymptotics for Packing and Covering on \(S^2\)

In this section, we use the solution to the best-packing and the best-covering problems in the plane to obtain the leading term of the best-packing distance and the best-covering radius on the two-dimensional Euclidean sphere.

Let \(\psi :\mathbb R^3\setminus \{{ 0}\}\rightarrow S^2\) be the radial projection onto \(S^2\); i.e., the mapping defined by \(\psi ({ x})=\frac{{ x}}{\left| { x}\right| }\), \({ x}\in \mathbb R^3\setminus \{{ 0}\}\). For every point \({ a}\in S^2\), we denote by \(P_{ a}\) the tangent plane to \(S^2\) at point a.

For arbitrary \(\epsilon >0\), let \(\delta =\delta _\epsilon >0\) be such that for every \({ a}\in S^2\),
$$\begin{aligned} (1+\epsilon )^{-1}\left| { x}-{ y}\right| \le \left| \psi ({ x})-\psi ({ y})\right| \le \left| { x}-{ y}\right| , \ \ { x},{ y}\in P_{ a}\cap B[{ a},\delta ]. \end{aligned}$$
(3.7.1)
We will use the following auxiliary construction.

Lemma 3.7.1

For every \(\delta >0\), there exists a partition \(\mathcal D_\delta \) of \(S^2\) into regions \(Q_1,\ldots , Q_l\) with the property that for every \(i=1,\ldots , l\), there is a point \({ q}_i\in Q_i\), such that the set \(K_i:=\psi ^{-1}(Q_i)\cap P_{{ q}_i}\) is a convex polygon of diameter less than \(\delta \) with at most six sides.

Proof

To prove the lemma we construct an example of such a partition \(\mathcal D_\delta \). We first partition \(S^2\) into 2m congruent subsets \(S_1,\ldots , S_{2m}\) by \(m>2\) vertical planes that contain the z-axis and form equal angles (of \(\frac{\pi }{m}\)). Each subset \(S_i\) is the intersection of \(S^2\) and the dihedral angle \(L_i\) formed by two neighboring half-planes. Let \(l_i\) be the line in the xy-plane that passes through the origin and is perpendicular to the half-plane bisecting the dihedral angle \(L_i\). Let \(M_i\) be the plane containing the z-axis and the line \(l_i\). We now partition the set \(S_i\) into 2m patches by \(2m-1\) half-planes whose common edge is the line \(l_i\) all of which are contained in the same half-space relative to the plane \(M_i\) as the set \(S_i\) with every two neighboring half-planes forming an angle of \(\frac{\pi }{2m}\) (one of these half-planes lies in the xy-plane).

The patches \(Q_{i}\), \(1\le i\le l=4m^2\), obtained in this way form a required partition \(\mathcal D_\delta \) if the positive integer m is sufficiently large. Indeed, pick some point \({ q}_{i}\) in the interior of every patch \(Q_{i}\). Since \(Q_{i}\) is the intersection of \(S^2\) with a polyhedral cone, which has three or four facets, the set \(K_{i}=\psi ^{-1}(Q_{i})\cap P_{{ q}_{i}}\) is a triangle or a convex quadrilateral. It only remains to pick m large enough so that \({{\,\mathrm{diam}\,}}K_{i}<\delta \), \(i=1,\ldots , l\).    \(\square \)

We are now ready to prove the main theorems of this section (see Chap.  12 for more general results concerning packing on rectifiable sets). The first result deals with best packing.

Theorem 3.7.2

Let \(\omega _N^*\), \(N\ge 2\), be an N-point best-packing configuration on \(S^2\) and let \(N_r\), \(r\in (0,2]\), be the largest number of points on \(S^2\) with minimal pairwise separation at least r. Then
$$\begin{aligned} \lim \limits _{N\rightarrow \infty }\delta (\omega _N^*)\sqrt{N}=\lim \limits _{r\rightarrow 0^+}{r\sqrt{N_r}}=2 \sqrt{2\pi }/\root 4 \of {3}. \end{aligned}$$
(3.7.2)

Proof

The first equality in (3.7.2) can be shown by a straightforward argument (we leave it to the reader) provided that the second equality is proved. So we deal with the second equality in (3.7.2).

Let \(\epsilon >0\) be arbitrary and \(\delta =\delta _\epsilon >0\) be defined by (3.7.1). For \(\mathcal D_\delta \) as in Lemma 3.7.1, set \(P_i:=P_{{ q}_i}\) and let \(\psi _i:P_{i}\rightarrow S^2\) be the restriction of the mapping \(\psi \) onto the plane \(P_{i}\), \(i=1,\ldots , l\). Then \(Q_i=\psi _i(K_i)\), where \(K_i:=\psi ^{-1}(Q_i)\cap P_{i}\).

To prove the upper bound, we pick a configuration \(X_r\), \(r\in (0,2]\), of \(N_r\) points on \(S^2\) such that \(\delta (X_r)\ge r\). Denote \(Y_i:=\psi _i^{-1}(X_r\cap Q_i)\) and let \(K_i^r\) be the polygon in the plane \(P_{i}\) that contains the r-neighborhood of \(K_i\) relative to \(P_i\) and is similar to \(K_i\) with the smallest possible coefficient. Clearly, \(Y_i\subset K_i^r\), \(\delta (Y_i)\ge \delta (X^r)/(1+\epsilon )\ge r/(1+\epsilon )\) and \(\mathcal L_2(K_i^r)\rightarrow \mathcal L_2(K_i)\), \(r\rightarrow 0^+\). The collection of open circles in \(P_{i}\) of radius \(r/(2(1+\epsilon ))\) centered at the points of \(Y_i\) is pairwise disjoint and is contained in \(K_i^r\), which is a convex polygon with at most six sides. By Lemma 3.4.3, we have \(\# Y_i\le \frac{4(1+\epsilon )^2}{r^2\sqrt{12}}\mathcal L_2(K_i^r)\), \(i=1,\ldots , l\). Thus
$$ N_r\le \sum \limits _{i=1}^{l}\#(X_r\cap Q_i)=\sum \limits _{i=1}^{l}\# Y_i\le \frac{4(1+\epsilon )^2}{r^2\sqrt{12}}\sum \limits _{i=1}^{l}\mathcal L_2(K_i^r), $$
and so
$$\begin{aligned} \limsup \limits _{r\rightarrow 0^+}r^2 N_r\le \frac{4(1+\epsilon )^2}{\sqrt{12}}\sum \limits _{i=1}^{l}\mathcal L_2(K_i)\le \frac{2(1+\epsilon )^4}{\sqrt{3}}\sum \limits _{i=1}^{l}\mathcal L_2(Q_i)= \frac{8\pi (1+\epsilon )^4}{\sqrt{3}}. \end{aligned}$$
(3.7.3)
To prove the lower bound we put \(\rho :=r(1+\epsilon )\), \(r\in (0,2]\), and let \(H_{i, r}\), \(i=1,\ldots , l\), be an equi-triangular lattice in the plane \(P_{i}\) with minimal pairwise separation \(\rho \). Define also \(T_{i,\rho }:=\{{ x}\in K_i : \mathrm{dist} ({ x},\partial K_i)>\rho \}\), where \(\partial K_i\) is the boundary of the polygon \(K_i\). Let \(V_{ x}\) be the Voronoi cell in the plane \(P_i\) of the point \({ x}\in H_{i,\rho }\) with respect to the lattice \(H_{i,\rho }\). Since the set \(T_{i, 2\rho }\) is covered by the collection of Voronoi cells \(V_{ x}\) with \({{ x}\in H_{i,\rho }\cap T_{i,\rho }}\) and \(\mathcal L_2(V_{ x})=\sqrt{3}\rho ^2/2\), we have \(\# (H_{i,\rho }\cap T_{i,\rho })\cdot \sqrt{3}\rho ^2/2\ge \mathcal L_2(T_{i, 2\rho })\). Now let \(Z_i:=\psi _i(H_{i,\rho }\cap T_{i,\rho })\subset S^2\) and \(A_r:=\cup _{i=1}^{l}Z_i\). Observe that \(\delta (Z_i)\ge r\) and that the Euclidean distance from any point \({ z}\in Z_i\) to the boundary of the set \(Q_i=\psi _i(K_i)\) relative to \(S^2\) is at least \(\rho /(1+\epsilon )=r\). Then the interior of the spherical cap of radius r centered at each \({ z}\in Z_i\) contains no points from sets \(Q_j\) with \(j\ne i\). Consequently, \(\delta (A_r)\ge r\) and we obtain
$$ N_r\ge \# A_r=\sum \limits _{i=1}^{l}\# Z_i=\sum \limits _{i=1}^{l}\# (H_{i,\rho }\cap T_{i,\rho })\ge \frac{2}{\rho ^2\sqrt{3}} \sum \limits _{i=1}^{l}\mathcal L_2(T_{i, 2\rho }). $$
Since \(\mathcal L_2(T_{i, 2\rho })\rightarrow \mathcal L_2(K_i)\), \(r\rightarrow 0^+\), in view of (3.7.1) we get
$$ \liminf \limits _{r\rightarrow 0^+}r^2N_r\ge \!\frac{2}{\sqrt{3}(1+\epsilon )^2}\sum _{i=1}^{l}\mathcal L_2(K_i)\!\ge \!\frac{2}{\sqrt{3}(1+\epsilon )^2}\sum _{i=1}^{l}\mathcal L_2(Q_i)\!=\!\frac{8\pi }{\sqrt{3}(1+\epsilon )^2}. $$
Taking into account (3.7.3) and letting \(\epsilon \rightarrow 0\), we deduce that \(\lim \limits _{r\rightarrow 0^+} r^2 N_r=8\pi /\sqrt{3}\), which implies the second equality in (3.7.2).    \(\square \)

The second result concerns optimal covering.

Theorem 3.7.3

Let \(\omega _N^c\), \(N\in \mathbb N\), be a best-covering N-point configuration on \(S^2\) and let \(M_r\), \(r\in (0,2]\), be the minimal number of points on \(S^2\) such that the union of closed balls of radius r centered at these points covers \(S^2\). Then
$$\begin{aligned} \lim \limits _{N\rightarrow \infty }{\eta (\omega _N^c, S^2)\sqrt{N}}=\lim \limits _{r\rightarrow 0^+}{r\sqrt{M_r}}=2\sqrt{2\pi }/\root 4 \of {27}. \end{aligned}$$
(3.7.4)

Proof

The first equality in (3.7.4) can be shown by a straightforward argument, which is left to the reader. To show the second equality, choose an arbitrary \(\epsilon >0\) and let \(\delta =\delta _\epsilon >0\) be defined by (3.7.1). Denote by \(D_\delta \) a partition of \(S^2\) from Lemma 3.7.1 into pairwise disjoint sets \(Q_1,\ldots , Q_l\). Let, as above, \(P_i=P_{{ q}_i}\) and define the mapping \(\psi _i:P_i\rightarrow S^2\) to be the restriction of the mapping \(\psi \) onto the plane \(P_i\). For every i, the polygon \(K_i=\psi _i^{-1}(Q_i)\) is the intersection of \(n_i\le 6\) closed half-planes \(H_1,\ldots , H_{n_i}\) determined by the lines \(l_1,\ldots , l_{n_i}\) containing the sides of \(K_i\). Let \(H_j'\) be the closed half-plane contained in \(H_j\) determined by the line \(m_j\) obtained by shifting \(l_j\) by a distance \(\rho >0\) into the half-plane \(H_j\). Denote by \(K_{i,\rho }\) the polygon obtained as the intersection of the half-planes \(H_1',\ldots , H_{n_i}'\).

Let \(X_r\), \(r\in (0,2]\), be a configuration of \(M_r\) points on \(S^2\) such that the union of closed balls of radius r centered at the points of \(X_r\) covers \(S^2\). Then \(\eta (X_r, S^2)\le r\). Let \(h_0>0\) be small enough so that \(X_r\cap Q_i\ne \emptyset \), \(i=1,\ldots , l\), \(r\in (0,h_0]\) and set \(\rho =r(1+\epsilon )\). The collection of circles \(\{B[{ y},\rho ]\}_{{ y}\in Y_i}\) in the plane \(P_i\), where \(Y_i:=\psi _i^{-1}\left( X_r\cap Q_{i}\right) \), covers the set \(K_{i,\rho }\), \(i=1,\ldots , l\). Indeed, every x in the interior of \(K_{i,\rho }\) is at a distance of more than \(\rho \) from the boundary of \(K_i\). Then the point \(\varphi ({ x})\in Q_i\) is at a distance greater than r from the boundary of \(Q_i\). There is \({ z}\in X_r\cap Q_i\) such that \(\left| \psi ({ x})-{ z}\right| \le r\), and in view of (3.7.1), \(\left| { x}-\psi ^{-1}_i({ z})\right| \le \rho \), where \(\psi _i^{-1}({ z})\in Y_i\). Since the disks are closed, their union covers the closure of the polygon \(K_{i,\rho }\).

The convex polygon \(K_{i,\rho }\) has at most six sides and by Lemma 3.4.4, we have \(\#Y_i\ge 2\mathcal L_2(K_{i,\rho })/(\rho ^2\sqrt{27})\). Thus
$$ r^2M_r=r^2\sum \limits _{i=1}^{l}\#Y_i\ge \frac{2}{(1+\epsilon )^2\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal L_2(K_{i,\rho }) $$
and so
$$\begin{aligned} \begin{aligned} \liminf \limits _{r\rightarrow 0^+}r^2M_r&\ge \frac{2}{(1+\epsilon )^2\sqrt{27}}\lim \limits _{r\rightarrow 0^+}\sum \limits _{i=1}^{l}\mathcal L_2(K_{i,\rho }) =\frac{2}{(1+\epsilon )^2\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal L_2(K_{i})\\&\ge \frac{2}{(1+\epsilon )^4\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal H_2(Q_{i})=\frac{8\pi }{(1+\epsilon )^4\sqrt{27}}. \end{aligned} \end{aligned}$$
Since \(\epsilon \) is arbitrary, \(\liminf \limits _{r\rightarrow 0^+}r^2M_r\ge 8\pi /\sqrt{27}\).
To establish the upper bound, let \(H^r:=\sqrt{3} {r}\varLambda ^*\), where we recall that \(\varLambda ^*\) is the equi-triangular lattice generated by vectors \({ v}_1=(1,0)\) and \({ v}_2=\left( 1/2,\sqrt{3}/2\right) \). Let \(K_i^r:=\{{ x}\in P_i : \mathrm{dist}({ x}, K_i)\le r\}\) and \(Z_i:=\psi (H^r\cap K_i^r)\). The Voronoi cells of the points in \(H^r\) are regular hexagons of area \(\sqrt{27}r^2/2\). Furthermore, the union of Voronoi cells of points from \(H^r\cap K_i^r\) is contained in \(K_i^{2r}\). Thus \(\# Z_i\le 2\mathcal L_2(K_i^{2r})/(\sqrt{27}r^2)\). The set \(Q_i\) is covered by the collection of disks \(\{B[{ y}, r]\}_{{ y}\in Z_i}\); indeed, for every \({ x}\in Q_i\), there is \({ z}\in H^r\cap K_i^r\) such that \(\left| \psi _i^{-1}({ x})-{ z}\right| \le r\). Then \(\left| { x}-\psi ({ z})\right| \le r\), where \(\psi ({ z})\in Z_i\), and so \(\eta (Z, S^2)\le r\), where \(Z=Z_1\cup \ldots \cup Z_l\). Consequently,
$$ M_r\le \# Z\le \sum \limits _{i=1}^{l}\# Z_i\le \frac{2}{r^2\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal L_2(K_i^{2r}), $$
and we obtain
$$ \limsup \limits _{r\rightarrow 0^+}{r^2M_r}\le \frac{2}{\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal L_2(K_i)\le \frac{2(1+\epsilon )^2}{\sqrt{27}}\sum \limits _{i=1}^{l}\mathcal L_2(Q_i)=\frac{8\pi (1+\epsilon )^2}{\sqrt{27}}. $$
Hence \(\limsup \limits _{r\rightarrow 0^+}{r^2M_r}\le 8\pi /\sqrt{27}\), which together with the lower bound implies the second equality in (3.7.4).    \(\square \)

3.8 Notes and Historical References

Section 3.1:

More information on the best-packing problem and extended reviews of known results can be found in the books by Fejes Tóth [131, 134], Rogers [233], Thompson [272], Conway and Sloane [92], Böröczky [40], and others. Proposition 3.1.2 can be found, for example, in the thesis by Borodachov [43] or in the work by Bondarenko, Hardin, and Saff [37].

Figure 3.1 was provided by Robert Womersley.

Section 3.2:

Theorems 3.2.6 and 3.2.9 as well as the example following Theorem 3.2.9 have been obtained by Bondarenko, Hardin, and Saff [37]. The proof of Proposition 3.2.7 follows a standard argument as in [37, Theorem 1]. Concerning the optimality of \(N=11\) points on \(S^2\), see the works by Danzer [99] and Böröczky [39].

Section 3.3:

Theorem 3.3.1 can be found, for example, in the book by Böröczky [40, Section 6.1]. Theorem 3.3.2 which provides the solution to the best-packing problem on \(S^2\) for \(N=5\) was proved by Tammes [269]. Corollary 3.3.3 is the result by Fejes Tóth [130]. Theorem 3.3.2 and Corollary 3.3.3 also follow from a more general result given in the book by Böröczky [40, Theorem 6.2.1] and in the earlier works listed in [40] at the end of Section 6.2. Theorem 3.3.4 and Corollary 3.3.10 were proved by Fejes Tóth [130, 131, 134]. Theorem 3.3.4 provides, in particular, best-packing configurations on \(S^2\) for \(N=4\) (see Theorem 3.3.1), \(N=6\) (see Corollary 3.3.3), and \(N=12\) (see Corollary 3.3.6). On the sphere \(S^2\) best-packing configurations are also known for \(7\le N\le 9\), see the work [244] by Schütte and van der Waerden. For \(N=10\) and 11 they are found in the paper by Danzer [99] with shorter proofs given by Hars [162] for \(N=10\) and Böröczky [39] for \(N=11\). The configuration of \(N=13\) points that solves the best packing problem on \(S^2\) was found by Musin and Tarasov [203] who also showed its uniqueness up to rotation. A best-packing configuration of \(N=14\) points on \(S^2\) was found by these authors as well [204]. The case \(N=24\) of the best-packing problem on \(S^2\) was solved by Robinson in [230]. Known results on best-packing on a higher dimensional sphere are discussed further in Chapter  5.

The result concerning the best-covering configuration of \(N=8\) points on \(S^3\) was proved by Dalla, Larman, Mani-Levitska, and Zong in [97]. The proof can be found in the book by Böröczky [40, Section 6.7]. The proof of Theorem 3.3.12 can be found in the work by Galiev [137] or in the book [40, Theorem 6.5.1]. Lemma 3.3.13 was proved in [40]. The case \(d=2\) of Theorem 3.3.14 was proved by Schütte [243] while the case \(d\ge 3\) was established by Böröczky and Wintsche [41].

Theorem 3.3.15 and its restatement, Theorem 3.3.18, were established by Fejes Tóth [131, 134]. They imply the solution to the optimal covering problem on \(S^2\) for \(N=4\) (see the case \(d=2\) of Theorem 3.3.12) and \(N=6,12\) (see Corollary 3.3.17). For \(N=5\) and \(N=7\), the optimal covering problem on \(S^2\) was solved by Schütte [243] while for \(N=8\), it was solved by Wimmer [289]. Finally, for \(N=10\) and \(N=14\), the solution was found by Fejes Tóth [128].

More information on the optimal-covering problem and more reviews of known results can be found in the books by Fejes Tóth [131, 134], Rogers [233], Conway and Sloane [92], Böröczky [40], and others.

Figures 3.2 and 3.3 were provided by Robert Womersley.

Section 3.4:

The result by Thue (without a rigorous proof) stating that \(\varDelta _2={\pi }/{\sqrt{12}}\) and that the packing of circles of radius 1 in \(\mathbb R^2\) centered at points of \(2\varLambda ^*\) has the highest density, see Theorem 3.4.5, was published in [275, 276]. Fejes Tóth’s proof of Thue’s theorem can be found in [129] (see also his book [131]). Rigorous proofs can also be found in the works by Segre and Mahler [247], Davenport [100], Hsiang [168], and Hales [149]. Concerning best-packing in \(\mathbb R^3\), Fejes Tóth in [131] suggested a plan, which was used by Hales and Ferguson together with a complex computer-aided proof, to show that \(\varDelta _3={\pi }/{\sqrt{18}}\) [150]. The proof has been fully validated by the Flyspeck team [270] in 2014, who gave a formal proof based on the Isabelle and HOL Light formal proof assistants. The face-centered cubic packing (FCC) and the hexagonal close packing (HCP) are the densest sphere packings in \(\mathbb R^3\). This resolved the Kepler conjecture which first appeared in [172].

Viazovska [283] proved that \(\varDelta _8=\pi ^4/384\) and that the sphere packing with centers at the \(E_8\)-lattice has the highest density. Cohn, Kumar, Miller, Radchenko, and Viazovska [86] proved that \(\varDelta _{24}=\pi ^{12}/(12!)\) and that the sphere packing with centers at the Leech lattice has the highest density.

The lattice packing problem in \(\mathbb R^p\) requires finding the densest packing of balls of equal radii whose centers form a full-rank lattice in \(\mathbb R^p\). This problem was solved by Gauss in \(\mathbb R^3\) in an Anzeige of a book by Ludwig August Seeber. The “face-centered cubic”  (FCC) packing is the unique densest sphere packing in \(\mathbb R^3\). The FCC packing consists of all vectors in \(\mathbb R^3\) with integer components such that exactly one or exactly three components are even. An interesting fact is that in the FCC packing every ball is touched by exactly 12 other balls at points which do not form a regular icosahedron. The densest lattice packing is also known in dimensions \(4\le p\le 8\) (see the book by Conway and Sloane [92] for references). In particular, in dimension 8, the densest lattice packing is the \(E_8\)-lattice sphere packing, which is known to be the densest among arbitrary packings (see the work by Viazovska [283]). Furthermore, in [85] Cohn and Kumar proved that the centers of the balls in the densest lattice packing in \(\mathbb R^{24}\) must form the Leech lattice. They also showed in that paper that no sphere packing in \(\mathbb R^{24}\) (lattice or non-lattice) can exceed the density of the Leech lattice packing by a factor of \(1+1.65\cdot 10^{-30}\). Later, Cohn, Kumar, Miller, Radchenko, and Viazovska [86] proved that the Leech lattice packing is the densest among all packings in \(\mathbb R^{24}\).

Regarding Hales’ result [150] that \(\varDelta _3=\pi /\sqrt{18}\), any packing consisting of “horizontal layers” of spheres with centers in an equi-triangular lattice such that every sphere touches exactly three spheres from the layer above and exactly three spheres from the layer below achieves this best packing density. Two such best packings are the face-centered cubic (FCC)  packing (whose centers form a lattice) and the hexagonal close packing  (HCP). In the HCP centers of balls in every “odd” (“even”) layer are placed above centers of balls in the previous “odd” (“even”) layer while in the FCC centers of balls in every layer are placed above the holes of the previous layer of the same parity.

Lemma 3.4.2 is proved using the Euler characteristic formula, see, e.g., the book by Trudeau [278]. Kershner’s result on best covering in \(\mathbb R^2\) (Theorem 3.4.6) can be found in [173]. For another proof, see [131].

Section 3.6:

The conjecture that the equality \(\varDelta _p=\varDelta _p^*\) fails for some dimension p can be found in the book by Conway and Sloane [92]. The statement of Theorem 3.6.8 (which implies Theorem 3.6.1 when \(\varOmega \) is a ball) was published by Minkowski [196, 197]. In [197] he proved it when the domain \(\varOmega \) is a ball but never published the proof of the general case. The first proof of the general case of Theorem 3.6.8 was published by Hlawka [163]. The proof presented in this book uses the ideas of Hlawka [163] and of Rogers who later gave a simpler proof of Theorem 3.6.8 in [231]. Lemmas 3.6.3 and 3.6.4 were proved by Rogers in [231]. Lemma 3.6.5 was proved in the papers by Hlawka [163] and Siegel [253]. The proof of Theorem 3.6.10 is due to Blichfeldt [33].

The lower and upper bounds for the highest sphere packing density \(\varDelta _p\) presented in Section 3.6 were later improved by a number of authors. The following table lists known lower bounds for the highest lattice sphere packing density \(\varDelta ^*_p\) (which are also lower bounds for \(\varDelta _p\)).

Lower bound for \(\varDelta ^*_p\)

Reference

\(2{\zeta (p)}2^{-p}\)

Minkowski [197]

\(((1/2)\log 2+o(1))p2^{-p}\)

Davenport and Rogers [102]

\(2(p-1)\zeta (p)2^{-p}\)

Ball [12]

\((6/e+o(1))p2^{-p}\)

Vance [281]

\(Cp2^{-p}\), \(C>6/e\),

Venkatesh [282]

\(cp \log \log p \ \! 2^{-p}\), \(c>0\), \(p\in \mathcal N\), \(\mathcal N\) is sparse

Venkatesh [282]

The table below lists known upper bounds for the highest sphere packing density \(\varDelta _p\).

Upper bound for \(\varDelta _p\)

Reference

\((p/2+1)2^{-p/2}\)

Blichfeldt [33]

\((p/e)2^{-p/2}\)

Rogers [232]

\(2^{-p(0.5096+o(1))}\)

Sidel’nikov [252]

\(2^{-p(0.5237+o(1))}\)

Levenshtein [188, 189]

\(2^{-p(0.599+o(1))} \)

Kabatjanskiĭ and Levenshtein [171]

Cohn and Zhao [89] have shown that the Kabatjanskiĭ and Levenshtein bound can be improved “on average with respect to p” by the factor 1.236....

Section 3.7:

The leading term of the best-packing distance and of the optimal-covering radius on a compact set of positive Lebesgue measure in \(\mathbb R^p\) with boundary of Lebesgue measure zero was obtained by Kolmogorov and Tihomirov in [175]. This result for the covering radius was later reproved by Graf and Luschgy in [144, Chapter II, Section 10]. Theorems 3.7.2 and 3.7.3 were proved by Habicht and van der Waerden (see [147] and [148]). These two theorems are also consequences of Theorems 3.3.4 and 3.3.15 proved by Fejes Tóth in [130, 131, 134].

Footnotes

  1. 1.

    See ( 2.7.1) for the definition.

  2. 2.

    A non-empty subset B of a metric space A is called path connected  if for any two points \(x, y\in B\), there is a continuous function (“path”) \(\gamma :[0,1]\rightarrow A\) such that \(\gamma (0)=x\), \(\gamma (1)=y\), and \(\gamma ([0,1])\subset B\).

  3. 3.

    Some authors define this ratio as \(2\eta (\omega _N, A)/\delta ^\rho (\omega _N)\).

  4. 4.

    The great circle equidistant from the antipodal points

  5. 5.

    Recall that, as introduced in Section  1.3, \(\mathcal {H}_2\) denotes 2-dimensional Hausdorff measure.

  6. 6.

    The definition of the \(E_8\) lattice is given in Subsection  5.8.3.

  7. 7.

    The definition of the Leech lattice is given in Subsection  5.8.4.

  8. 8.

    In fact, we have \(U_{a,q}=q\mathbb Z^p \cup D_{a, q}\); however, for our proof we only need a one-sided inclusion.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Sergiy V. Borodachov
    • 1
    Email author
  • Douglas P. Hardin
    • 2
  • Edward B. Saff
    • 2
  1. 1.Department of MathematicsTowson UniversityTowsonUSA
  2. 2.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA

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